The differential equation d^2y/dx^2 - 2 dy/dx + y = x^2 has complementary function y = Ae^x +...

Question:

Second-Order Non-Homogeneous Equations:

A second order differential equation can be written in general as,

{eq}\frac{d^2y}{dx^2}+p_1(x)\frac{dy}{dx}+p_2(x)y=f(x) {/eq},

its general solution has the form,

{eq}y=\psi(x,A,B) {/eq}, where two constants are needed to satisfy the initial conditions for {eq}dy/dx {/eq} and {eq}y {/eq}.

When the coefficients {eq}p_1(x) {/eq} and {eq}p_2(x) {/eq} are constant the equation reduces to,

{eq}\frac{d^2y}{dx^2}+p_1\frac{dy}{dx}+p_2y=f(x) \qquad \qquad (1) {/eq}.

The solution strategy for the constant coefficients second order differential equation consists of two steps:

1- Determining the solution of the homogeneous equation by proposing an exponential ansatz {eq}y(x)=e^{\lambda x} {/eq} and solving the resulting characteristic equation, {eq}\lambda^2+p_1\lambda+p_2=0 {/eq}. The solution is built as a linear combination of the exponential for the roots of the characteristic equation ie: {eq}y_h(x)=C_1e^{\lambda_1x}+_2e^{\lambda_2x} {/eq}.

2- Finding a particular solution, {eq}y_p(x) {/eq}, of the inhomogeneous differential equation following the undetermined coefficients method. An ansatz is proposed for {eq}y_p(x) {/eq} such that it has the same functional form of {eq}f(x) {/eq}, after substituting in (1) we solve for the coefficients resulting in the desired particular solution.

The complete solution for equation (1) is {eq}y(x)=y_h(x)+y_p(x) {/eq}.

Physics is full of second order differential equations, both with constant and non-constant coefficients. Lets show a few examples:

i) {eq}\frac{d^2x}{dt^2}+\frac{\gamma}{m}\frac{dx}{dt}+\frac{k}{m}x=\frac{F\cos{\omega_0 t}}{m} {/eq}, the equation of the damped oscillator under the action of an external force is a second order differential equation with constant coefficients.

ii) {eq}-\frac{\hbar^2}{2m}\frac{d^2 \Psi(x)}{dx^2}+(V(x)-E)\Psi(x)=0 {/eq}, the stationary Schrodinger equation is also a second order differential equation whose coefficients can be either constant or non constant depending on the form of the potential {eq}V(x) {/eq}. In general is a very complex problem for which no general solution strategy can be described.

Answer and Explanation: