Dynamical Systems Modelling Part I: Fundamentals
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Authors: Denis Mulumba and Yusuf Brima
Introduction to physical/dynamical process modelling
In our day to day lives, a lot of natural and artificial phenomena can be described and/ or explained using Mathematical tools. Many of these phenomena change with time, some in a more sophisticated way compared to others. This inspires the modelling of such systems termed to as Dynamical Systems. Dynamical Systems can range from simple interaction of predators and preys in a habitat, for example the variation of the population of foxes and rabbits, to viral spread (such as COVID-19) in a population. The set of mathematical tools used to model such systems allow us, we mean, any one with a fair bit of motivation and access of a modern computer programming environment, to model and make meaningful predictions of the time-evolving properties if such systems. That's why, modelling techniques of dynamical systems are extensively used in engineering, economics, epidemiology, physics and many more disciplines.
However, the factors and variables determine the cumbersomeness of the dynamical system under consideration. These systems' behavior can be interpreted to the re-known differential equations from which further examinations can be done to well explain how these systems behave.
Differential Equations (Ordinary and Partial)
If you are a numbers nerd or have an inclination to mathematics, odds are, you may have come across this linchpin of modern mathematics. If not, be rest assured, because you are about to learn something profound which will change your view of the world around you. If you are wondering, we are not exaggerating! Thus, in this blog, we want to journey with you into the amazing beauty and power of this approach to model time-varying systems. So, what is a differential equation? A differential equation (often abbreviated DE) can be defined as an equation that contains one or more functions of one independent variable and their corresponding derivatives. So, if you are acquainted with functions and and high school calculus (i.e., differential and integral), we are in a better footing if not, don't worry. Learn the basics of functions and derivatives and come back and continue. Differential equations range from ordinary to partial. The generic form of a differential equation is thus:
\[\begin{equation} \frac{d \mathbf{x}}{d t}=f(\mathbf{x}), \mathbf{x} \in \mathbb{R}^{d}, \label{eq:differential_equation} \end{equation}\]with initial value $\mathbf{x}\left(t_{0}\right)=\mathbf{x}_{0}$. As mentioned above, ordinary differential equation (ODE) is one of types of DEs. It is called ordinary because it entails a $\textbf{single independent variable}$ like $\textbf{x}$ in equation $\ref{eq:ode_sample}$.
\[\begin{equation} \frac{dy}{dx}=f(x). \label{eq:ode_sample} \end{equation}\]An ODE does not impose the relationship between partial derivatives of a multivariate function as it is for a partial differential equation (PDE). What is a multivariate function? One may wonder, what is a multivariate function anyway?! A multivariate (otherwise known as multivariable) function is just a fancy name for a function that has multiple inputs: if you are a programmer, think of it as a function with multiple arguments/parameters and multiple return a value. That's it, you get the idea. A mathematical notational representation is shown in equation $\ref{multi}$. In contrast, a function with single input and a single output is called a single-variable function. Think of, for example a black-box, which you enter in and your height (in meters for example) comes out on the other side as shown in equation $\ref{single}$.
\[\begin{equation} f: \mathbb{R}^{n} \rightarrow \mathbb{R}. \label{multi} \end{equation}\]As example of a multivariate function is shown as follows:
\[\begin{equation} f(x, y)=x^{3}-y^{3}+3 x y+7, \end{equation}\]which is a function of two independent variables $x$ and $y$.
\[\begin{equation} f: \mathbb{R} \rightarrow \mathbb{R}. \label{single} \end{equation}\]An example of a single variable function can be as follows:
\[\begin{equation} f(x)=x^{2} + 2, \end{equation}\]where the function has $x$ only as the independent variable. The example of a single variable function above is a stated oversimplification, because in reality, it could be linear, non-linear, cubic, quadratic or polynomial in form. However, that is the basis of all ODE formulations. It has one independent variable and it's derivatives. So you may be wondering, are there nuts and bolts about ODEs worth pointing out. You definitely guessed it! There are, in fact a slew of of technical details to which becomes are whole area of specialization in the study of differential equations. So, to be just to the field, we shall brush through the main (we mean what we deem to the bare essentials) ideas. To home these ideas, let's take a simple example of an ODE:
\[\begin{equation} y+\frac{d y}{d x}=5 x, \end{equation}\]the equation above is a function of $y$ with respect to (w.r.t for short in mathematical speak) the variable $x$ and it's derivative $\frac{d y}{d x}$. The derivative is the key to it been called a differential equation as shown above. So pay a closer attention to this piece of fact as we proceed and that's what make this mathematical form of abstraction very powerful. The $y$ is called a function, because in maths, a dependent variable is function of its input(s) plus any other constant(s). You may be wondering why on earth we are talking about the power (and beauty) of these mathematical artifacts. The reason is simple! In our universe (and maybe others beyond our current scientific knowledge), things change, and to be able to describe this change is what led our 17th Century ancestors: Leibniz, the Bernoulli brothers and others to conjure the ideas of differential equations.
To deeply appreciate the usefulness of DEs, let's talk about rabbits albeit in math terms. The intuition about the evolutionary behaviour of rabbit population in a suitable habitat is pretty straightforward -- at least if you have some environmental priors, which is not rocket science. Put bluntly, the more rabbits we have the more baby rabbits they can make over time. There is an inherent feedback loop effect because those baby rabbits will grow and have more babies too. The cycle continues as the population grows faster and faster until some limit (physical as you would imagine like running out of resources) is reached. So to put this into a mathematical framework would be as follows: at any given time $t$, we will have $N$ rabbits with a growth rate $r$. Therefore, the population's rate of change is:
\[\begin{equation} \frac{dN}{dt} \label{rabbits} \end{equation}\]If you have some fair bit of calculus under your belt, then it will become apparent equation \ref{rabbits} is a derivative (or slope as some would call it). It means $\lim\limits_{t \to +\infty}$, we want to capture the extent to which the population has changed. If we unpack the heavy notation, we want to measure the minutest change in the population of rabbits. A time smaller than a zeptosecond (i.e., one trillionth of a billionth of a second). If we suppose that the growth rate $r$ is 0.01 new rabbits per week for every current rabbit population $N$. This means when $N$ is 1000, the rate of change $\frac{dN}{dt}$ is then $1000 \times 0.01 = 10$ new rabbits per week. However, the drawback to our formulation is it is true for only that population instance and doesn't account for the change that is happening continuously. But one lesson we can learn is the bigger the population, the more new rabbits there can be! This implies if the population is 2000 we get $2000\times0.01 = 20$ new rabbits per week and so on. So a more rigorous formulation of our rabbit population model would be as follows:
\[\begin{equation} \frac{dN}{dt} = rN. \label{rabit_fin} \end{equation}\]PDE models are used together with specific boundary conditions and initial conditions to best describe the system's behavior. These PDEs such as the Navier-Stokes equations are useful in many aspects to study the dynamical behavior of many physical phenomena such as the movement of fluids that is most discussed in fluid dynamics, a key component of climate physics. On the other hand, Ordinary differential equations such as:
- Lorenz’s equations are used in examination of complicated problems and examining the butterfly effects.
- The Lotka-Volterra equations that are very much useful in the examination of the prey-predator behavior which is a an important concept for co-existence.
- Differential equation such the Susceptible Infected Recovered (SIR)-model, Susceptible Exposed Infected Recovered (SEIR)- models that are used in the dynamical modelling of diseases, such as Covid-$19$ are important in vaccine development and prediction of the disease spread.
- For mechanical engineering, the concept of ODEs is dominant in modelling motion such as oscillations of a swinging pendulum drawing from damped to un-damped oscillations.
These and many other examples of differential equations, both ODEs and PDEs are proving to be useful to computer scientist, physicists, climate scientists and many other scientists that contribute to the study of dynamical behaviors of all physical aspects on this planet.
Order and Degree of Differential Equations
Order of a differential equation
The order of a differential equation is the highest power on the variable in any differential equation. The order of any differential equation determines the number of arbitrary constants needed to be added to the general solution.
The degree of a differential equation
The degree of a differential equation is the highest power of the highest order derivative in the differential equation. This is determined after the equation has been cleared from fractions and all radicals as far as the derivatives are concerned.
Some illustrative examples
Consider a differential equation
\[\begin{align} \frac{d^3y}{dx^3} + \bigg(\frac{d^2y}{dx^2}\bigg)^4 - \bigg(\frac{dy}{dx}\bigg)^{-4}+y^2+ 4 = 0 \label{aaa} \end{align}\]From equation $\ref{aaa}$, the highest power of the derivative is $3$. This is the order of this equation. However, this is not the degree of this equation. The degree of this equation is $1$ since the highest order derivative $ \frac{d^3y}{dx^3}$ is indexed with a unity. Following this analogy, we have with no doubt that you can think of the order and degree of the following differential equations.
\[\begin{equation*} \bigg(\frac{d^6y}{dx^6}\bigg)^2+ 4\bigg(\frac{d^2y}{dx^2}\bigg)^6+4y=0 \end{equation*}\] \[\begin{equation*} \frac{d^4y}{dx^4}+ 4\bigg(\frac{d^2y}{dx^2}\bigg)^2-\frac{dy}{dx}-4y=10 \end{equation*}\] \[\begin{equation*} \bigg[\frac{d^2y}{dx^2}+1 \bigg]=a\frac{d^2y}{dx^2} \end{equation*}\] \[\begin{equation*} y'' + (y''')^3 = (x+y''')^3 \end{equation*}\]Let's talk about notation
A long time ago, clever apes (our progeny), figured out that lifting all the numbers in their head was a great mental gymnastic and hassle. Secondly, communicating ideas, mostly in written form was cumbersome if not a a dead-end! What did they resort to? Not surprisingly, symbols. Oh yes! Symbols. So you many be wondering, why the detour on symbols. It's because symbols (the basis for notations) have a significant role to play in mathematics. Different mathematicians have their "taste" and they even have disciples to that end. Back to differential equations. If you are new to this province of the exclusive few, then calculus in general (to be specific differentiation) has the following notations:
The Leibniz notation was originally employed by Gottfried Leibniz which is the main stay of mathematics. Mathematicians love it because it has a higher explanatory power. This notation is very suitable when the equation $y = f(x)$ is regarded as a functional relationship between dependent and independent variables $y$ and $x$. Using this notation, we can express the derivative as follow:
\[\begin{equation} \frac{dy}{dx}\text{.} \label{eq:lib1} \end{equation}\]Equation \ref{eq:lib1} is the first-order derivative of $y = f(x)$. It called first-order because it is the first time we are differentiate $y$ w.r.t. $x$. The function whose value evaluated at $x$ is the derivative of $f$ at $x$ can be written as
\[\begin{equation*} \frac{df}{dx}(x) \text{ or } \frac{d f(x)}{d x} \text{ or } \frac{d}{d x} f(x)\text{.} \end{equation*}\]With the Leibniz notation, higher order derivatives can be written as
\[\begin{equation*} \frac{d^{2} y}{d x^{2}}, \frac{d^{3} y}{d x^{3}}, \frac{d^{4} y}{d x^{4}}, \ldots, \frac{d^{n} y}{d x^{n}}\text{.} \end{equation*}\]Another important notation worth looking at is the Lagrange's notation. It is also common among mathematicians and scientists and it's named after the Italian mathematician and astronomer Joseph Louis Lagrange. However, it was reportedly invented by Euler and just popularized by the Lagrange. Using Lagrange's notation, a prime mark ($\prime$) denotes a derivative. For a given function $f$, then its derivative evaluated at $x$ is written as follow:
\[\begin{equation*} f^{\prime}(x)\text{.} \end{equation*}\]The second derivative is written as $f^{\prime \prime}(x)$ while the third derivative is as $f^{\prime \prime \prime}(x)$ but we run into trouble moving past that point. Higher order derivatives $4 \dots $ often use either Hindu-Arabic or Roman numerals as shown below.
\[\begin{equation*} f^{(4)}(x), f^{(5)}(x), f^{(6)}(x), \ldots, \end{equation*}\]or
\[\begin{equation*} f^{\mathrm{iv}}(x), f^{\mathrm{v}}(x), f^{\mathrm{vi}}(x), \ldots \end{equation*}\]We also have Newton's notation for differentiation sometimes referred to as the dot notation ($\dot{}$) or jokingly the flyspeck notation. The key idea using this notation is to place a dot over the dependent variable. This means, suppose $y$ is the dependent variable that is a function of $t$, then the derivative of $y$ w.r.t. $t$ is
\[\begin{equation*} \dot{y}\text{.} \end{equation*}\]The second and third derivatives can be represented using multiple dots, as in
\[\begin{equation*} \ddot{y}, \dddot{y} \end{equation*}\]Newton this idea to the next level for higher order derivatives:
\[\begin{equation*} \ddot{y}, \ddot{y},\dddot{y} \stackrel{4}{\dot{y}}, \ldots, \stackrel{n}{\dot{y}} \end{equation*}\]If this couldn't whet your appetite, you can quench your curiosity by learning more from Wikipedia and later continue. Now that you're abreast of the bells and whistles, let's continue.
Why dynamical systems?
The motivation behind the study of dynamical systems draws from their diverse field of applications. These range from Electrical engineering, electronics, computer engineering, bioinfomatics, climate science and many other aspects of science. The theory of dynamical systems describes general patterns found in the solutions of the systems of linear equations. The beauty of using this is that even non-linear equations can be linearized using mathematical tools and the behavior studied on a linearized ground. The typical dynamic variable is often time, if it is the only dynamic variable, the analysis is then based on an ODE and PDE modes are used with geometrical considerations are put in place in addition to time. For this blog post, we shall focus on the theory that focus on those equations representing only the change of processes in time.
Close-form analytical solutions
The nature of a differential equation one looks at determines the technique one can use to analytically solve it. This section clearly discusses some of the mostly used techniques when analytically solving ordinary differential equations and also explicitly discusses the significant difference between ODEs and PDEs.
Some illustrative examples
Consider a differential equation
\[\begin{align} \frac{dy}{dx} = f(y)g(x)\label{a} \end{align}\]The function of $y$, ($f$) in equation (\ref{a}) can be put on the left hand side of this equation through a simple mathematical tool of division. Likewise, the term $dx$ can be brought to the right hand side. The resulting equation then clearly has terms in $y$ and $x$ on different side.
\[\begin{align*} f(y)dy = g(x)dx \end{align*}\]If this is attained, the we say that these two variables, $x$ and $y$ are separated. If this is attained, we can apply integration techniques to this equation to find an analytical solution. The solution then becomes;
\[\begin{align*} \int f(y) dy = \int g(x)dx +c, \end{align*}\]were $c$ is the constant of integration. We can therefore say that to to find an analytical solution to the differential equation using separation of variables, the following steps are key.
- Move all the $y$ terms including $dy$ to one side of the equation and all those in $x$ including $dx$ to the other side.
- Integrate one side with respect to $y$ and the other side with respect to $x$ without forgetting the constant of integration.
- We can the simply to obtain the required analytical solution.
These steps therefore imply that this technique cannot be used to differential equations where step one cannot be attained.
Note: It should be noted that $x$ and $y$ represent the independent and dependent variables respectively. These represent any variables in a given differential equation. A quick thought can be the equation for population growth, One can define $N$ as the population at a particular time, $t$. In this case, the independent and dependent variables are $t$ and $N$ respectively.
Some illustrative examples
If one thinks about the rate of heat transfer from a cup of hot tea, $y$ being proportional to the quantity of heat present at that time, $t$. Then, we can write
\[\begin{align*} \frac{dy}{dt} \propto y \end{align*}\]This then simplifies to
\[\begin{align} \frac{dy}{dx} = -ky \label{aa} \end{align}\]where $k$ is the proportionality constant. The negative attached to this constants implies that heat drops with time (cooling). Applying the first step of separating variables will mean that $\ref{aa}$ is
\[\begin{align*} \frac{1}{y}dy = -kdt \end{align*}\]Step two then yields
\[\begin{align*} \ln{y} = -k + C \end{align*}\]The last step then will give the analytical solution to this differential equation by separation of variables as;
\[\begin{align*} y = A \exp{^{-kt}} \end{align*}\]where $A = e^c$, a constant.
First order linear equations
Consider a differential equation
\[\begin{align*} \frac{dx}{dt} = f(t,x) \end{align*}\]If the function $f$ is a sum of terms which are either independent of $x$ or linear in $x$, then the equation is said to be linear. we are now be concerned about how to solve a differential equation of such a form, for example:
\[\begin{align} \frac{dx}{dt} + 4tx = t \label{z} \end{align}\]From our definition of a Linear first order DE, we note that equation $\ref{z}$ is a typical example of these. These take on the form
\[\begin{align} \frac{dy}{dx} + P(x)y = Q(x)\label{zz} \end{align}\]To deal with such, we introduce a strong mathematical expression known as the integrating factor.
We define the integrating factor, $\mu (x)$ as
\[\begin{align*} \mu (x) = e^{\int P(x) dx} \end{align*}\]Then, using this, we can transform $\ref{zz}$ into a separable differential equation (that is always friendly in terms of solving) by multiplying it to this equation. Precisely, we have
\[\begin{align*} e^{\int P(x) dx} \frac{dy}{dx} + e^{\int P(x) dx} P(x) y = e^{\int P(x) dx} Q(x) \end{align*}\]The left hand side of this equation is then an exact equation that can simply be written as;
\[\begin{align*} \frac{d}{dx}(\mu(x) P(x)) = \mu (x) Q(x) \end{align*}\]Using this technique to solve equation $\ref{z}$ we therefore has the integrating factor as
\[\begin{align*} \mu (t) =& e^{\int 4t dt}\\ =& e^{2t^2} \end{align*}\]Using this results to ensure exactness on equation implies that
\[\begin{align*} e^{2t^2} \frac{dx}{dt} + 4te^{2t^2}x = t e^{2t^2} \end{align*}\]We can therefore smile that exactness is assured on the left hand side of this equation and so confidently write this as
\[\begin{align*} \frac{d}{dt}(xe^{2t^2}) = t e^{2t^2} \end{align*}\]which is separable. Simplification of this has been discussed in the previous subsection aligning with separation of variables and the final solution to this can be written with techniques of integration such as substitution.
\[\begin{align*} xe^{2t^2} = \frac{1}{4}e^{2t^2} +C \end{align*}\]And this simplifies to
\[\begin{align*} x(t) = \frac{1}{4} +Ce^{-2t^2}. \end{align*}\]Exact Equations
At this point, many of us might be a little scared about some terms used in the previous section! Its no surprise that some of us are in this state. It is good that this comes out of curiosity. Let's use this section to over turn the concept of Exact Equations. To get this right, let us start by considering the following equation
\[\begin{align} (x -\sin y)\frac{dy}{dx}+y = 0\label{m} \end{align}\]We note that we can beautify $\ref{m}$ by simply doing some expansion and this yields
\[\begin{align*} (x\frac{dy}{dx}+y) - \sin y \frac{dy}{dx} = 0 \end{align*}\]It can clearly be observed that
\[\begin{align*} x \frac{dy}{dx} + y = \frac{d}{dx}(xy) \end{align*}\]and
\[\begin{align*} -\sin y \frac{dy}{dx} = \frac{d}{dx}(\cos y) \end{align*}\]Putting that on hold, let us define $\psi (x,y) = xy +\cos y$. We can therefore write $\ref{m}$ as
\[\begin{align*} \frac{d}{dx}(\psi(x,y(x)) = 0 \end{align*}\]And if we integrate this equation after separation of variables gives
\[\begin{align*} \psi (x, y(x)) = C; \hspace{0.5cm} \text{where C is a constant of integration} \end{align*}\]Equations that can be written in this form for some function $\psi$ are termed to as Exact. A very big question that is not answered yet concerns how to determine exactness. Well, let's consider the differential equation below
\[\begin{align} N(x,y) \frac{dy}{dx}(x) + M(x,y) = 0 \label{mm} \end{align}\]where $N$ and $M$ are differentiable functions $N,M : \mathbb{R}^2 \to \mathbb{R}$ If these conditions are satisfied, $\ref{mm}$ is therefore an Exact deifferential equation. For any twice differentiable function, we can write $\psi : \mathbb{R}^2 \to \mathbb{R}$. By the chain rule,
\[\begin{align*} \frac{d}{dy}\psi(x,y(x)) = \frac{\partial \psi}{\partial y } \frac{dy}{dx}(x) + \frac{\partial \psi}{\partial x} \end{align*}\]So, we can precisely say that $\ref{mm}$ is twice differentaible if there exists a function $\psi : \mathbb{R}^2 \to \mathbb{R}$ with
\[\begin{align*} \frac{\partial \psi}{\partial y}(x,y) = N(x,y), \end{align*}\]and
\[\begin{align*} \frac{\partial \psi}{\partial x} (x,y)= M(x,y). \end{align*}\]Also, since $\psi$ is twice differentiable, then
\[\begin{align*} \frac{\partial ^2 \psi}{\partial x \partial y} = \frac{\partial ^2 \psi}{\partial y \partial x}. \end{align*}\]We note that this function ($\psi$) exists if
\[\begin{align*} \frac{\partial N}{ \partial x} = \frac{\partial M}{\partial y}, \end{align*}\]With this background information, we take a critical look at our illustrative example $\ref{m}$.
Clearly, this is exact since
\[\begin{align*} \frac{\partial }{\partial x} (x - \sin y) = 1 = \frac{\partial }{\partial x} (y). \end{align*}\]To find the function, $\psi$, we need to let
\[\begin{align*} \frac{\partial }{\partial y} = (x - \sin y) \hspace{0.2cm} \text{and} \hspace{0.2cm} \frac{\partial }{\partial x} = y. \end{align*}\]This implies that
\[\begin{align} \psi(x,y) = xy - \cos y + \phi (x) \label{n} \end{align}\]The term $\phi$ in equation (\ref{n}) is a function of $x$ and is a result of integrating the multivariate function $\psi$ with respect to $y$. Differentiating this with respect to $x$ gives
\[\begin{align*} \frac{\partial \psi}{\partial x} = y + \phi ' (x) \end{align*}\]We already have an equation in terms of $ \frac{\partial \psi}{\partial x} $. If we use this in finding the value of the function independent of $y$, $\phi$, implies that
\[\begin{align*} \phi ' (x) = 0 \hspace{0.2cm} \iff \phi = C, \end{align*}\]where $C$ is the constant of integration. We can therefore deduce that, the general solution to the differential equation is
\[\begin{align*} \psi (x,y) = xy - \cos y + C. \end{align*}\]Changing Variables
Some ODEs can be easily solved in a more simplified way by simply changing variables in the given equation. ODEs of the form
\[\begin{align} \frac{dy}{dx} = f\bigg(\frac{y}{x}\bigg), \label{aaw} \end{align}\]are termed to as Homogeneous equations . A change of variable can be applied to simply reduce the 'scare' caused when the equation under consideration proves to be sophisticated.
If we let
\[\begin{equation*} u = \frac{y}{x}, \end{equation*}\]our goal is to form a new simplified ODE independent of the variable $y$ but rather in terms of the new variable introduced above.
This therefore implies that
Once this is attained, we can always run to the given equation and form a simplified version of the form.
Example
Give that we are required to find the general solution to the differential equation below
\[\begin{align} \frac{dy}{dx} = \frac{xy}{x^2+y^2}\label{aw} \end{align}\]Observe that we can write \((\ref{aw})\) in the form of \((\ref{aaw})\) as
\[\begin{align*} \frac{dy}{dx} = \frac{(\frac{y}{x})}{(1+ \frac{y}{x})^2}. \end{align*}\]So, we have a homogeneous equation and we can therefore apply change of variable techniques.
Letting
we have
\[\begin{align*} x \frac{du}{dx} + u = \frac{u}{1+u^2}. \end{align*}\]This simplifies to,
\[\begin{align*} x \frac{du}{dx} = -\frac{u^3}{1+u^2}. \end{align*}\]This can be easily solved using techniques of change of variables $[To be continued]$
Bernoulli Equation
Equations of the form
\[\begin{align} \frac{dy}{dx} + \alpha(x) y = \beta(x) y ^{\gamma}, \label{c} \end{align}\]where $\gamma \in \mathbb{R} - {0,1}$ are termed to as Bernoulli Equations. To solve equations like ($c$), we note that this equation can be written as
\[\begin{align*} y^{-\gamma} \frac{dy}{dx} + \alpha(x) y^{1-\gamma} = \beta(x) \end{align*}\]If we let
\[\begin{align*} u = y^{1-\gamma}, \end{align*}\]it implies that
\[\begin{align*} \frac{du}{dx} = (1- \gamma) y ^{-\gamma}\frac{dy}{dx}. \end{align*}\]Therefore, we can write $(\ref{c})$ independent of the terms in $y$. That is
\[\begin{align*} \frac{1}{1-\gamma} \frac{du}{dx} + \alpha(x) u = \beta (x) \end{align*}\]which is a linear first order ODE that can be solved using techniques discussed above.
An quick example that we wish to discuss here is; If we consider a differential equation below;
Observe that $(\ref{wan})$ takes on the form of $(\ref{c})$. Therefore, dealing with this will simply call for the above discussion treatments of such equations. That is to say
divide this equation through by $y^4$. This gives
Then, if we let
\[\begin{equation*} u = y^{-3}, \end{equation*}\]we have that
\[\begin{align*} \frac{dy}{dx} = - \frac{1}{3}y^4 \frac{du}{dx}. \end{align*}\]The differential equation taking on the form of Bernoulli's equations can therefore be written in terms of the new dependent variable, $u$, and independent of the former, $y$, as
\[\begin{align*} -\frac{1}{3}\frac{du}{dx} + u = 1, \end{align*}\]which can be written in the form
\[\begin{align*} \frac{dy}{dx} + p(x)y = Q(x). \end{align*}\] Let us not forge that such equations become a simple 'sip' or easy like taking a cup of tea in the morning when one employs the above discussed integrating factor.
For this problem specifically, the integrating factor associated to this is
We are happy that we have previously discussed the treatment of such equation and therefore confidently state that following the above steps, one can arrive to the simplified general solution in terms of $u$ as
\[\begin{align*} u = 1 + C e^{3x} \end{align*}\]Well, we need to always remember that the solution we seek is in terms of variable $y$. Therefore, if we substitute for $u$, we shall have
\[\begin{align*} y(x) = \frac{1}{\bigg(1+C e^{3x} \bigg)^3} \end{align*}\]As the general solution to this differential equation.
Second Order Equation
Aside ordinary differential equations of order $1$ discussed above, together with techniques that can be applied to them in search for right general solutions, differential equations of order $2$ are discussed in this blog.
Consider a differential equation of the form
\[\begin{align} a \frac{d^2y}{dx^2} + b \frac{dy}{dx} + c y = d(x). \label{q} \end{align}\] In $(\ref{q})$, $a, b, c$ and $d$ can either be constants of functions in terms of the given variables.
We should also note that if the right hand side vanishes, the differential equation is then termed to as homogeneous, otherwise it is non-homogeneous. Different techniques can be employed in looking for general solutions to second order ODE depending on the cumbersomeness of the equation. Discussed below are some of the most common techniques.
Great job! You may be interested in reading part II of this blog post series where we delve into the numerical approximation methods and their general applications to dynamical systems modelling.
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