How to solve the ODE y'' + y = sin(x)

[toastie]

Homework Statement

y"+y=sinx

initial conditions: y(90 deg) = 3, y(45 deg) = 2

Calculate y at x = -1

Homework Equations

y=u+v

The Attempt at a Solution

I have gotten the following:

r^2 + 1 = 0 Therefore, r1=i and r2 = -i

u=(c1)cosx + (c2)sinx

v=Asinx + Bcosx
v'=Acosx - Bsinx
v"=-Asinx - Bcosx

-Asinx -Bcosx + Asinx + Bcosx - sinx = 0

everything cancels down to: -sinx = 0. Thus, v=0

Then I get,

y=(c1)cosx + (c2)sinx + 0

y(45 deg) shows that c2=2

y(90 deg) shows that c1=3

Thus, y=3cosx + 2sinx

y(-1) = -.06204

Could someone please let me know where I am making a mistake?

Thank you in advance!

 

[g_edgar]

The solution of the homogeneous case has terms cos x and sin x in it. The right-hand-side *also* has one of those terms in it. Presumably your textbook has the information of what to do in this exceptional case... as you found, the usual method does not work.