General solution of second order ODE

[smart_worker]

Homework Statement

Find the general solution.

Homework Equations

y"+y=x²sin2x

The Attempt at a Solution

Characteristic equation would be:
m² + 1 = 0

So,m² = -1

Therefore, m = i or m = -i.

Complementary function would be : Asinx+Bcosx where,A and B are constants respectively.

If I write the particular integral as (Cx²+Dx+E)sin2x + (Px²+Qx+R)cos2x
Then,it would be very tedious to solve.

Is there any alternative way like writing sin2x as Imaginary part of e^i2x and then solving the particular integral?

 

[Mark44]

Homework Statement

Find the general solution.

Homework Equations

y"+y=x²sin2x

The Attempt at a Solution

Characteristic equation would be:
m² + 1 = 0

So,m² = -1

Therefore, m = i or m = -i.

Complementary function would be : Asinx+Bcosx where,A and B are constants respectively.

If I write the particular integral as (Cx²+Dx+E)sin2x + (Px²+Qx+R)cos2x
Then,it would be very tedious to solve.

But not that tedious. All you have to do is take the first and second derivatives, and substitute them into your DE to determine the six constants. It might be there is another way, but this is how I would do the problem.

Is there any alternative way like writing sin2x as Imaginary part of e^i2x and then solving the particular integral?

 

[rude man]

Another approach is Laplace transform:

For the right-hand term,
if f(x) ↔ F(s), then
x²f(x) ↔ F''(s)
and f(x) = sin(2x) and F(s) = L{sin(2x)}.

You can incorporate the two initial conditions on y also in the usual manner if they're not identically zero.