Answer to Question #289440 in Differential Equations for Pankaj

Question

Using the method of undetermined coefficients, solve the equation:

d²y/dx² -3dy/dx +2y=4x²

Answer 1

$\frac{d²y}{dx²} - 3 \frac{dy}{dx} + 2y=4x²\\ \Big\{\frac{d²}{dx²} - 3 \frac{d}{dx} + 2\Big\}y=4x²\\ \text{Let} \, m = \frac{d}{dx} \\ \text{Then, we have the auxiliary equation}\\ m^2 - 3m + 2 = 0 \\ m^2 - m - 2m + 2 = 0 \\ m(m - 1) - 2(m - 1) = 0\\ (m - 2)(m - 1) = 0\\ m = 2,1 \\ \text{Hence, we have the complementary function}\\ C.F = C_1 e^x + C_2 e^{2x} \\ \text{where} \, C_1 \text{and} \, C_2 \, \text{are arbitrary constants} \\ \text{To find the particular integeral by}\\ \text{the method of undetermined coefficient}\\ \text{Let the particular integral be of the form}\\ y_p = ax^2 + bx + c \\ \text{We want to find a, b, c such that}\\ y''_p - 3y'_p + 2 y_p = 4x^2 . . . (*) \\ \text{From} \, y_p = ax^2 + bx + c \, \text{.We have}\\ y'p = 2ax + b \\ y''p = 2a \\ \text{Now, substitute these values into eq(*)}\\ \text{ To have,}\\ 2a - 3(2ax + b) + 2(ax^2 + bx + c) = 4x^2 \\ \text{Comparing powers. We have}\\ 2ax^2 = 4x^2 \, \implies \, 2a = 4 \, \implies a = 2\\ (-6a + 2b)x = 0x \, \implies -12 + 2b = 0\\ \implies b = 6\\ 2a - 3b + 2c = 0 \, \implies 4 - 18 + 2c = 0\\ \implies 2c = 14 \implies c = 7\\ \text{Hence, }\\ \text{P.I}\, = y_p = ax^2 + bx + c = 2x^2 + 6x + 7\\ \text{Thus, the general solution is}\\ y(x) = C.F + P.I \\ y(x) = C_1 e^x + C_2 e^{2x} + 2x^2 + 6x + 7 \\ \text{where} \, C_1 \, \text{and} \, C_2 \, \text{are arbitrary constants}$