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

Find the general solution of y''+3y'+2y=12x2.

1. y = C1 e- x + C2 e- 2 x + 6 x2 + 18 x + 21
2. y = C1 e- x + C2 e- 2 x + 6 x2 - 18 x + 21
3. y = C1 e- x + C2 e- 2 x + 6 x2 - 18 x + 9
4. y = C1 e- x + C2 e- 2 x + 6 x2 - 18 x

Answer

B. y = C1 e- x + C2 e- 2 x + 6 x2 - 18 x + 21

Solution

Complementary: r=-1,-2. Try yp=Ax2+Bx+C: 2A+3(2Ax+B)+2(Ax2+Bx+C)=12x2. Matching: 2A=12 A=6; 6A+2B=0 B=-18; 2A+3B+2C=0 C=21.

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