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

  • #1

Homework Statement


Solve ODE
[tex]y''-y=e^{-t}[/tex]

[tex]y(0)=1, y'(0)=0[/tex]


Homework Equations





The Attempt at a Solution


Homogenuous solution

[tex]t^2-1=0[/tex]

[tex]y=C_1e^t+C_2e^{-t}[/tex]

From

[tex]y(0)=1, y'(0)=0[/tex]

[tex]y=\frac{1}{2}e^t+\frac{1}{2}e^{-t}[/tex]

How from that get complete solution?
 

Answers and Replies

  • #2
hunt_mat
Homework Helper
1,741
25


It's wrong. What you have to do it write:
[tex]
y=C_{1}e^{t}+C_{2}e^{-t}
[/tex]
and then find the particular integral, call it [itex]f(x)[/itex] say, and then apply the boundary condition to the function:
[tex]
y=C_{1}e^{t}+C_{2}e^{-t}+f(x)
[/tex]
 
  • #3


How to find particular integral?
 
  • #4
hunt_mat
Homework Helper
1,741
25


I would look for a function
[tex]
y=Ate^{-t}
[/tex]
and likewise.
 
  • #5


How do you know how to look for the function?
 
  • #6


How you choose form of particular solution?
 
  • #7
I like Serena
Homework Helper
6,577
176

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