Ordinary differential equaiton

[sara_87]

Homework Statement

dy/dt = -y + cos(pi*t)

Homework Equations

The Attempt at a Solution

first, i took the y to the other side and then found an integraing factor to be e^t;
multiplied the ODE by e^t then integrated both sides wrt t. i have the initial condition
y(2)=4
so my general solution is:
y = {(pi*sin(pi*t) + (cos(pi*t))/[pi(pi+1)] } + {e^t(4-1/(pi(pi+1)))}

Is this correct?

 

[SirOtis]

I end up getting c = 4e^2 and y = (1/(pi*e^t))sin(pi*t)+4e^(2-t) When you solve for c the sin term should become 0 and you just multiply over the e^2. This is my first post, but I hope it helps.

 

[lurflurf]

gey in a habit of checking your solution in the equation
if
y = {(pi*sin(pi*t) + (cos(pi*t))/[pi(pi+1)] } + {e^t(4-1/(pi(pi+1)))}
is
dy/dt = -y + cos(pi*t)
with
y(2)=4

 

[sara_87]

I end up getting c = 4e^2 and y = (1/(pi*e^t))sin(pi*t)+4e^(2-t) When you solve for c the sin term should become 0 and you just multiply over the e^2. This is my first post, but I hope it helps.

how did you get to that?
For c, i got (4e^2)-(e^2/pi(pi+1))

 

[DavidWhitbeck]

I got

$$y(t) = \frac{(\cos \pi t + \pi \sin \pi t)}{\pi^2+1} + e^{2 - t}\left ( 4 - \frac{1}{\pi^2 + 1}} \right )$$

And I checked it with the DE and IC and everything seems to be in order.

 

[SirOtis]

Oh, I'm sorry. I think David is right. I forgot to multiply the right side by the integrating factor before integrating.