Solving Second Order ODE: y''-y=e^{-t} - Homework Solution

[matematikuvol]

Homework Statement

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

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

Homework Equations

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]

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:
$$y=C_{1}e^{t}+C_{2}e^{-t}+f(x)$$

 

[matematikuvol]

How to find particular integral?

 

[hunt_mat]

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

 

[matematikuvol]

How do you know how to look for the function?

 

[matematikuvol]

How you choose form of particular solution?

 

[I like Serena]

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