General solution of second order ODE

  • #1
smart_worker
131
1

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 ei2x and then solving the particular integral?
 

Answers and Replies

  • #2
36,653
8,652

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.
smart_worker said:
Is there any alternative way like writing sin2x as Imaginary part of ei2x and then solving the particular integral?
 
  • #3
rude man
Homework Helper
Insights Author
Gold Member
8,031
867
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.
 

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