Differential Equations

Many natural laws in Physics, Chemistry, and Engineering are expressed by means of differential equations. Among many natural laws, we treat Newton's 2nd law, Newton's cooling law, Kirchhoff's 2nd law, Fourier's law of heat transfer, wave equation, chemical equation.

A differential equation is a relation between an unknown function and derivatives of the unknown function. Differential equations we treat are the ones that can be solved by quadrature.

A relation among an independent variable $x$, and a function $y(x)$, $y^{\prime},y^{\prime \prime},\ldots,y^{(n)}$ expressed in

$$F(x,y(x),y^{\prime}(x),\ldots,y^{(n)}(x)) = 0$$

is called the ordinary differential equation in $y$.

We give some examples of differential equations. Note that $${y^{\prime} \equiv \frac{dy}{dx}, y^{\prime\prime} = \frac{d^{2}y}{dx^{2}}. }$$

$$\begin{array}{lll} (a) y^{\prime} = x^{3} &(b) y^{\prime}... ...e} = x & (f) xy^{\prime\prime} - \sin{y} = x^{3} \end{array}$$

The order of the differential equation is the highest order derivative present in the differential equation.

Differential equations $(a),(b),(d)$ are the 1st order and the rest is 2nd order. The natural laws we mentioned above are formulated as the 1st order or the 2nd order differential equation. In this book, we deal mainly with the 1st and the 2nd order differential equations.

A differential equation is called linear if the equation is expressed as an unknown function and it's derivatives. Otherwise it is called non linear. Thus, $n$th order linear differential equation is expressed in the following form.

$$a_{n}y^{(n)} + a_{n-1}y^{(n-1)} + \cdots + a_{1}y^{\prime} + a_{0}y = b(x),$$

NOTE From the definition above, differential equations $(a)-(c)$ are linear, $(d)-(f)$ are non linear.

If a function saitisfies the given differential equation on some interval, then the function is called the solution of the differential equation. To find a solution of the given differential equation, we say solve the differential equation. A solution may not be in the form of $y = f(x)$. It sometimes given by implicit function such as $G(x,y) = 0$.

Example 1..1   Show the functions $e^{2x},ce^{2x}$ are solutions of $y^{\prime} = 2y$. Show the functions $e^{x},e^{x}+c \ (c \neq 0)$ are not solution of $y^{\prime} = 2y$.

SOLUTION The derivatives of $e^{2x}$ and $ce^{2x}$ are $2e^{2x}$ and $2ce^{2x}$ respectively. Thus these satisfy $y^{\prime} = 2y$. Since $(e^{x})^{\prime} = e^{x} \not\equiv 2e^{x}, (e^{x}+c)^{\prime} = e^{x} \not\equiv 2e^{x}+c \ (c \neq 0)$. Thus they are not the solutions of $y^{\prime} = 2y$ $\ \blacksquare$

Example 1..2   Show that $y_{1} = x^{2} + x^{3}$ and $y_{2} = x^{3}$ are solutions of $y^{\prime} = 2(y/x) + x^{2}$.

SOLUTION $y_{1}^{\prime} = 2x + 3x^{2}$ and $2(y_{1}/x) + x^{2} = (2x^{2} + x^{3})/x + x^{2} = 2x + 3x^{2}$. Thus, $y_{1}(x)$ is a solution. Similarly, $y_{2}^{\prime} = 3x^{2}$ and $2(y_{2}/x) + x^{2} = 2x^{3}/x + x^{2} = 3x^{2}$. Thus, $y_{2}(x)$ is a solution $\ \blacksquare$

As you can see from this example, solutions of a differential equation may not be unique. In general, any $n$th order differential equation has $n$ different solutions.

We say a linear combination of $n$ independent solutions of $n$ constants the general solution.

Any solution derived from substituting a particular value into the general solution is called the particular solution. For example, $y = ce^{x}$ is the general solution of $y^{\prime} = y$ and $y = e^x$ is the particular solution.

A singular solution is a solution which can not be derived from substituting a value into the general solution.

Example 1..3   Show that $\frac{y}{1-y} = ce^x$ is the general solution of $y' = y(1-y)$. Then show no matter what value you choose for $c$, $y = 1$ can not be obtained. Show $y = 1$ is the singular solution.

SOLUTION Differentiate $\frac{y}{1-y} = ce^x$ with respect to $x$. Then

$$\frac{1}{(1-y)^2}y' = ce^x$$

Since $ce^x = \frac{y}{1-y}$, we have

$$\frac{1}{(1-y)^2}y' = \frac{y}{1-y}$$

Then

$$y' = \frac{y}{1-y}(1-y)^2 = y(1-y)$$

Thus, $\frac{y}{1-y} = ce^x$ is the general solution. Now $y = 1$ can not be derived from the general solution. Thus $y = 1$ is the singular solution.

Note that the general solution of a differential equation forms curves. A each curve is called the solution curve or the integral curve.

In a differential equation $F(x,y,y^{\prime},\cdots,y^{(n)}) = 0$, initial conditions are values of the solution and its derivatives at specific points such as

$$y(x_{0}) = y_{0}, y^{\prime}(x_{0}) = y_{1},\ldots,y^{(n-1)}(x_{0}) = y_{n-1}$$

The problem of finding a solution to the differential equation with the initial condition is called the initial value problem

In a differential equation $F(x,y,y^{\prime},\cdots,y^{(n)}) = 0$ on $[a,b]$, the condition must be satisfied by the solution or the derivatives of the solution is called the boundary conditions. The problem of finding a solution to the differential equation with the boundary conditions is called the boundary value problem