Engineering Board Exam Reviewer · Nonhomogeneous Second-Order Equations · hard

Find the general solution of y''-4y'+4y=e2x.

1. y = x2 e2 x2 + ≤ft(C1 + C2 x ) e2 x
2. y = x e2 x2 + ≤ft(C1 + C2 x ) e2 x
3. y = x2 e2 x + ≤ft(C1 + C2 x ) e2 x
4. y = x2 e2 x4 + ≤ft(C1 + C2 x ) e2 x

Answer

A. y = x2 e2 x2 + ≤ft(C1 + C2 x ) e2 x

Solution

Complementary: r2-4r+4=(r-2)2=0, so yc=(C1+C2x)e2x. Both e2x and xe2x are in yc, so try yp=Ax2e2x. Then yp''-4yp'+4yp=2Ae2x=e2x A= 12.

More Nonhomogeneous Second-Order Equations questions

1. Find the general solution of y''+y= x.
2. Solve y''+y=2 with y(0)=0 and y'(0)=0, and find y(π).
3. Find the general solution of y''+3y'+2y=12x2.
4. Find the general solution of y''-2y'-3y=6e-x.
5. Find the general solution of y''+4y=8e2x.
6. Find the general solution of y''-3y'+2y=4x.
7. Find the general solution of y''+y'-2y=6.
8. Find the general solution of y''-y=4.
9. Use variation of parameters to solve y''+y= x for 0<x< π2.

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