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


The differential equation $$\frac{\mathrm{d}^2 y}{\mathrm{d}x^2} - 2 \frac{\mathrm{d}y}{\mathrm{d}x} + y = x^2 $$ has complementary function {eq}y = Ae^x + Bxe^x {/eq}.

Find a particular integral of the differential equation.

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:

Become a member to unlock this answer! Create your account

View this answer

The homogeneous solution is provided in the exercise, notice that since the characteristic equation has a double root the homogenous solution includes...

See full answer below.

Learn more about this topic:

Undetermined Coefficients: Method & Examples
Undetermined Coefficients: Method & Examples


Chapter 10 / Lesson 15

The method of undetermined coefficients is used to solve a class of nonhomogeneous second order differential equations. This method makes use of the characteristic equation of the corresponding homogeneous differential equation.

Related to this Question

Explore our homework questions and answers library