# Solve ODE y''-y=e^{-t}

## Homework Statement

Solve ODE
$$y''-y=e^{-t}$$

$$y(0)=1, y'(0)=0$$

## The Attempt at a Solution

Homogenuous solution

$$t^2-1=0$$

$$y=C_1e^t+C_2e^{-t}$$

From

$$y(0)=1, y'(0)=0$$

$$y=\frac{1}{2}e^t+\frac{1}{2}e^{-t}$$

How from that get complete solution?

hunt_mat
Homework Helper

It's wrong. What you have to do it write:
$$y=C_{1}e^{t}+C_{2}e^{-t}$$
and then find the particular integral, call it $f(x)$ say, and then apply the boundary condition to the function:
$$y=C_{1}e^{t}+C_{2}e^{-t}+f(x)$$

How to find particular integral?

hunt_mat
Homework Helper

I would look for a function
$$y=Ate^{-t}$$
and likewise.

How do you know how to look for the function?

How you choose form of particular solution?

I like Serena
Homework Helper

Hi matematikuvol!

It is called the method of undetermined coefficients.
You can find it in wikipedia, although not quite in the form you need:
http://en.wikipedia.org/wiki/Undetermined_coefficients

Here's a better definition (just posted by another HH! ):

As an alternative you could use the method of Variation of parameters:
http://en.wikipedia.org/wiki/Variation_of_parameters