Question

Solution of $\frac {d^2y} {dx^2}=x\,e^x$ is y

• A. $e^x(x - 2) + C_1 x + C_2$
• B. $e^x(x - 1) + C_1 x^2 + C_2x$
• C. $e^x(x + 1) + C_1 x^2 + C_2x$
• D. $e^x(x - 2) - C_1 x + C_2$

Answer 1

Answer: (A)

$e^x(x - 2) + C_1 x + C_2$

Answer 2

We have, $\frac{d^{2}y}{dx^{2}} = xe^{x}$ Integrating $\left(i\right)$ by parts, we get $\frac{dy}{dx} = xe^{x} - \int e^{x}\,dx+C_{1} = xe^{x} - e^{x} + C_{1}$ Again integrating by parts, we get $y = xe^{x} - e^{x} - e^{x} + C_{1}x + C_{2}$ $\Rightarrow y = xe^{x} - 2e^{x} + C_{1}x+C_{2}$ $\Rightarrow y = e^{x} \left(x - 2\right) + C_{1}x + C_{2}$.