# Thread: Second order differential equations!

1. ## Second order differential equations!

Find y in terms of x given that
d2y/dx2 - 4(dy/dx) + 4y = e^2x

and that dy/dx=1 and y=0 at x=0

I've worked out that the complementary function is (A+Bx)e^2x

However, i don't understand why I should be using y=k(x^2)e^2x as the particular integral and not y=ke^2x

Can anyone explain?

2. Originally Posted by Erghhh
Find y in terms of x given that
d2y/dx2 - 4(dy/dx) + 4y = e^2x

and that dy/dx=1 and y=0 at x=0

I've worked out that the complementary function is (A+Bx)e^2x

However, i don't understand why I should be using y=k(x^2)e^2x as the particular integral and not y=ke^2x

Can anyone explain?
Since your complimentary solution contains a $\displaystyle xe^{2x}$ term, then by reduction of order, your particular solution must contain a $\displaystyle x^2e^{2x}$ term. Thus, I believe your particular solution must take on the form $\displaystyle y_p=\left(A_1+A_2x+A_3x^2\right)e^{2x}$.

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# d^2y/dx^2-4dy/dx 4y=2e^2x

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