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# Characteristic equation & imposing conditions watch

1. To find particular solution
y'' + 2y' + y = x^2 + 2

conditions y(0) = 3 and y(0) = 1

Since this a in-homogeneous equation do I just solve the RHS seperately x^2 and 2?
Would I need to express finding the particular function?

y'' + 2y' + y = 0

the characteristic equation has repeated roots at r = -1

So the homogeneous solution is: yh = K1 * e-x + K2 * x * e-x

The particular solution is: yp = A * x2 + B*x + C
2. Look for a particular integral of the form Ax^2+Bx+C.
3. (Original post by DFranklin)
Look for a particular integral of the form Ax^2+Bx+C.
Using these into original ODE we get

A * x2 + (4*A + B)*x + (2*A + 2*B + C) = x2 + 2

Solve for A, B & C

y = yh + yp = K1 * e-x + K2 * x * e-x + A * x2 + B*x + C

is that correct?
4. Yes

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