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

Find the general solution of y''-2y'-3y=6e-x.

  1. y = C1 e3 x + C2 e- x - 3 x e- x2
  2. y = C1 e3 x + C2 e- x + 3 x e- x2
  3. y = C1 e3 x + C2 e- x - 3 e- x2
  4. y = C1 e3 x + C2 e- x - 3 x2 e- x2

Answer

A. y = C1 e3 x + C2 e- x - 3 x e- x2

Solution

Complementary: r2-2r-3=(r-3)(r+1)=0, so yc=C1e3x+C2e-x. Since e-x already appears in yc, try yp=Axe-x. Then yp''-2yp'-3yp=-4Ae-x=6e-x A=- 32.

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