Chapter 4 Differential Equations
An equation containing differential coefficients is known as differential equation. The differential equation containing only one independent variable is known as ordinary differential equation. The differential equation containing more than one independent variable is known as partial differential equation. The changing entity is a variable and the rate at which one variable changes with respect to another is its derivative or differential coefficient. The differential coefficient or derivative of \(y [=f(x)]\) with respect to ’x’ (for any particular value of \(x\)) is defined as \(f'(x)\) or \(\frac{\,dy}{\,dx}\) such that
\begin{equation*}
f'(x) = \frac{\,dy}{\,dx} = \lim\limits_{\delta x \to 0}\frac{f(x+\delta x)-f(x)}{\,\delta x},
\end{equation*}
where \(x\) is independent variable and \(y\) is dependent variable which depends upon \(x\text{.}\) Here are some types of differential equations and their significance: Ordinary Differential Equations (ODEs): These equations involve derivatives with respect to a single independent variable. ODEs are essential in modeling dynamic systems, such as population growth, mechanical vibrations, and electrical circuits. Partial Differential Equations (PDEs): These equations involve derivatives with respect to multiple independent variables. PDEs are widely used in physics, engineering, and applied mathematics to describe phenomena like heat conduction, wave propagation, fluid flow, and quantum mechanics. Some examples of ordinary differential equations are
- \begin{equation*} \frac{\,dy}{\,dx} =\frac{1}{x}, \end{equation*}
- \begin{equation*} \frac{\,d^{2}y}{\,dx^{2}}+4\frac{\,dy}{\,dx}-y =0 \end{equation*}
- \begin{equation*} \left[1+\left(\frac{\,dy}{\,dx}\right)^{2}\right]^{3}=\left(\frac{\,d^{2}y}{\,dx^{2}}\right)^{2} \end{equation*}
- If z is a function of two independent variables x and y i.e., \(z=f(x,y)\) then \(\frac{\,dz}{\,dx}\) and \(\frac{\,dz}{\,dy} \) is partial derivatives of z with respect to x and y respectively. Hence the differential equation containing these partial derivatives are the partial differential equation, e.g.,\begin{equation*} \frac{\,dz}{\,dx}+\frac{\,dz}{\,dy} =x^{2}. \end{equation*}
Here, we confine in ordinary differential equation. The order of a differential equation is the order of the highest derivative appearing in the order of above lists, e.g., in list (1) order is 1 (i.e. first order) and that of (2) and (3) is 2 (i.e. second order). The degree of a differential equation is the highest power of the highest order derivative in the lists. The degree of lists (1) and (2) are 1 (one) and that of list (3) is 2.
If the dependent variable (say, y) and its derivatives appeared in the differential equation are of the first degree then the differential equation is known as linear differential equation, e.g., \(x\frac{dy}{dx}+y =e^{x}\) is a linear differential equation of \(1^{st}\) order and \(1^{st}\) degree. The differential equation is non-linear if it contains the product of dependent variable and its derivative or higher power of dependent variable or its derivative, e.g., \(\frac{dy}{dx}+xy^{2} =1\) is a non-linear differential equation as it contains \(y^{2}\text{.}\)
A standard form of a linear differential equation is given as
\begin{equation}
f_{o}(x)\frac{d^{n}y}{dx^{n}} +f_{1}(x)\frac{d^{n-1}y}{dx^{n-1}}+\cdots+f_{n-1}(x)\frac{dy}{dx}+ f_{n}(x)y = F(x)\tag{4.0.1}
\end{equation}
where \(f_{o}(x){,} f_{1}(x){,} f_{n}(x)\text{,}\) and \(F(x)\) are functions of independent variable x and \(f_{o}(x) \neq 0\text{.}\) When \(F(x)\) is zero, the equation is said to be homogeneous because each of its terms contains \(y(x)\) or a derivative. If \(F(x) \neq 0\text{,}\) then the equation is said to be inhomogenous, because some of its terms do not include \(y(x)\text{,}\) e.g., \(x^{2}\frac{\,dy}{\,dx}-2xy=\frac{1}{x}\text{.}\) The term \(\frac{1}{x}\) does not contain \(y\) hence the inhomogenous.
For Python code visit
https://apmonitor.com/pdc/index.php/Main/SolveDifferentialEquations
