Aspire Faculty ID #17161 · Topic: AMU MCA 2020 · Just now
AMU MCA 2020

The general solution of the differential equation $(D^2 - a^2)^3 y = e^{ax}$, where $D=\frac{d}{dx}$ is

Solution

Characteristic equation:

$(m^2-a^2)^3=0$

Roots: $m=\pm a$ each of multiplicity 3.

Hence complementary function:

$(c_1+c_2x+c_3x^2)e^{ax}+(c_4+c_5x+c_6x^2)e^{-ax}$

Since RHS is $e^{ax}$ and $m=a$ has multiplicity 3,

Particular integral = $x^3\frac{e^{ax}}{3!\,(2a)^3} =\frac{x^2e^{ax}}{8a^2}$

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