Question

Verify that xy = a e^x + b e^−x + x² is a solution of the differential equation \[x\frac{d^2 y}{d x^2} + 2\frac{dy}{dx} - xy + x^2 - 2 = 0.\]

Answer 1

We have,

\[xy = a e^x + b e^{- x} + x^2 \]

Differentiating with respect to x on both sides, we get

\[ \Rightarrow x\frac{dy}{dx} + y = a e^x - b e^{- x} + 2x\]

Again differentiating with respect to x on both sides, we get

\[ \Rightarrow x\frac{d^2 y}{d x^2} + \frac{dy}{dx} + \frac{dy}{dx} = a e^x + b e^{- x} + 2\]

\[ \Rightarrow x\frac{d^2 y}{d x^2} + 2\frac{dy}{dx} = xy - x^2 + 2 .........\left[ \because xy = a e^x + b e^{- x} + x^2 \right]\]

\[ \Rightarrow x\frac{d^2 y}{d x^2} + 2\frac{dy}{dx}- xy + x^2 - 2=0\]

Thus, xy = a e^x + b e^−x + x² is the solution of the given differential equation.