1. ## ODEs

Solve the differential equation
d2y/dx2 − 2dy/dx+ (B^2 + 1)y = e^x sin^2 x
for general values of the real parameter B.
Explain why this solution fails for B = 0
and B= 2 and find solutions for these values of B

i can find the complementary function but not the PI. how should i got about it?thanks

Solve the differential equation
d2y/dx2 − 2dy/dx+ (B^2 + 1)y = e^x sin^2 x
for general values of the real parameter B.
Explain why this solution fails for B = 0
and B= 2 and find solutions for these values of B

i can find the complementary function but not the PI. how should i got about it?thanks
The homogenous is,
$\displaystyle y''-2y'+(B^2+1)y=0$
The charachteristic equation is,
$\displaystyle k^2-2k+(B^2+1)k=0$
$\displaystyle k=1\pm \sqrt{1 - (B^2+1)} = 1\pm \sqrt{-B^2} = 1\pm i|B|$
So if $\displaystyle B=0$ then there is only one solution, if $\displaystyle B\not = 0$ then the quadradic has two solutions.

3. ## h

sure but what about the particular integral? ie the e^x sin^2x bit?