Differential Equations
An equation involving derivatives of one or more dependent variables with respect to one or more independent variables is called a differential equation. For example:
\begin{equation} \frac{d^{2}y}{dx^2} + xy \left( \frac{dy}{dx}\right)^2 = 0 \end{equation} \begin{equation} \frac{d^{4}x}{dt^4} + 5 \frac{d^{2}x}{dt^2} + 3x = \sin t \end{equation} \begin{equation}\frac{\partial v}{\partial s} + \frac{\partial v}{\partial t} = v \end{equation} \begin{equation}\frac{\partial^{2}u}{\partial x^2} + \frac{\partial^{2}u}{\partial y^2} + \frac{\partial^{2}u}{\partial z^2} = 0 \end{equation}
Ordinary Differential Equations
A differential equation involving ordinary derivatives of one or more dependent variables with respect to a single independent variable is called an ordinary differential equation. For example, +equation 1 and 2 are ordinary differential equations. In equation 1 the variable x is the single independent variable and y is a dependent variable and in equation 2 the independent variable is t, whereas x is dependent.
Partial Differential Equations
A differential equation involving partial derivatives of one or more dependent variables with respect to one or more independent variables is called a partial differential equation. Here, equation 3 and 4 are partial differential equations. In equation 3, the variables s and t are independent variables and v is the dependent variable. In equation 4 there are three independent variables: x, y, and z, in the equation u is dependent.
Order
The order of the highest ordered derivative involved in a differential equation is called the order of the differential equation. The order of equation 1 is of second order since the highest derivative involved is a second derivative. Equation 2 is an ordinary differential equation of the fourth order. The partial differential 3 and 4 are of the first and second orders, respectfully.
Linearity
A linear ordinary differential equation of order n, in the dependent variable y and the independent variable x, is an equation that is in, or can be expressed in, the form
\begin{equation} a_0(x)\frac{d^{n}y}{dx^{n}} + a_1(x)\frac{d^{n-1}y}{dx^{n-1}} + … + a_{n-1}(x)\frac{dy}{dx} + a_n(x)y = b(x) \end{equation}
where \(a_0\) is not identically zero.
Following ordinary differential equations are both linear.
\begin{equation} \frac{d^{2}y}{dx^2} + 5 \frac{dy}{dx} + 6y=0\end{equation} \begin{equation} \frac{d^{4}y}{dx^4} + x^2 \frac{d^{3}y}{dx^3} + x^3\frac{dy}{dx}=x e^{x} \end{equation}
In each case, y is the dependent variable. Observe that y and its various derivatives occur to the first degree only and that no products of y and/or any of its derivatives are present.
A nonlinear ordinary differential equation is an ordinary differential equation that is not linear. The following ordinary differential equations are all nonlinear:
\begin{equation}\begin{aligned}\frac{d^{2}y}{dx^2}+5\frac{dy}{dx}+6y^2 = 0 \\ \frac{d^{2}y}{dx^2}+5\left(\frac{dy}{dx}\right)^3+6y = 0 \\
\frac{d^{2}y}{dx^2}+5y\left(\frac{dy}{dx}\right)^3+6y = 0 \end{aligned}\end{equation}
Linear differential equations are further classified according to the nature of the coefficients of the dependent variables and their derivatives. For example, equation 6 is said to be linear with constant coefficients, while equation 6 is linear with variable coefficients.
Explicit vs Implicit
In mathematics, the explicit function is a function in which the dependent variable has been given “explicitly” in terms of the independent variable. Or it is a function in which the dependent variable is expressed in terms of some independent variables.
Let’s define an nth-order ordinary differential equation
\begin{equation} F\left[ x,y, \frac{dy}{dx},…,\frac{d^{n}y}{dx^n}\right] = 0,\end{equation}
where F is a real function of its (n+2) arguments \(x,y,\frac{dy}{dx},…,\frac{d^{n}y}{dx^n}.\)
Let \(f\) be a real function defined for all x in a real interval I and having an nth derivative for all \(x \in I\). The function \(f\) is called an explicit solution of the differential equation (9) on I if it fulfills the following:
$$F[x, f(x), f^{‘}(x),…,f^{(n)}(x)]$$ is defined for all \(x \in I,\) and
$$F[x, f(x), f^{‘}(x),…,f^{(n)}(x)] = 0$$ for all \(x \in I.\) For example
Let the function \(f\) defined for all real x by $$ f(x) = 2\sin x + 3 \cos x$$ is an explicit solution of the differential equation $$\frac{d^{2}y}{dx^2} + y = 0$$ for all real x.
Again, a relation \(g(x,y) = 0\) is called an implicit solution of (9) if this relation defines at least one real function \(f\) of the variable x on an interval I such that this function is an explicit solution of (9) on this interval. For example
The relation \begin{equation}x^2 + y^2 – 25 = 0\end{equation} is an implicit solution of the differential equation \begin{equation}x+y\frac{dy}{dx} = 0\end{equation} on the interval I defined by \(-5<x<5.\) For the relation (10) defines the two real functions \(f_1\) and \(f_2\) given by $$f_1(x) = \sqrt{25 – x^2}$$ and $$f_2(x) = -\sqrt{25 – x^2}.$$
respectively, for all real \(x \in I,\) and both of these functions are explicit solutions of the differential equation (11) on \(I\).