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Solution of the D.E y’’ + 3y’ + 2y = 12x2 when solved using the method of undetermined coefficients is ________

Solution of the D.E y’’ + 3y’ + 2y = 12x2 when solved using the method of undetermined coefficients is ________ Correct Answer y = c1 e – x + c2 e – 2x + 21 – 18x + 6x2

We have (D2 + 3D + 2)y = 12x2 A.E is m2 + 3m + 2 = 0 –> (m + 1)(m + 2) = 0 –> m = – 1, – 2 yc = c1 e – x + c2 e – 2x and ∅(x) = 12x2 and 0 is not a root of the A.E, we assume P.I in the form yp = a + bx + cx2….(1)to find a,b & c such that yp’’ + 3yp’ + 2yp = 12x2….(2), yp‘ = b + 2cx, yp” = 2c now (2) becomes 2c + 3(b + 2cx) + 2(a + bx + cx2) = 12x2 (2a + 3b + 2c) + (2b + 6c)x + (2c)x2 = 12x2 2a + 3b + 2c = 0, 2b + 6c = 0, 2c = 12 –> c = 6, b = – 18, a = 21 hence (1) becomes yp = 21 – 18x + 6x2 thus complete solution is y = yc + yp –> c1 e – x + c2 e – 2x + 21 – 18x + 6x2.

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