If $f(x)$ and $g(x)$ are two solutions of the differential equation a $\frac{d^2 y}{d x^2}+x^2 \frac{d y}{d x}+y=e^x$, then $f(x)-g(x)$ is the solution of
Answer & explanation
Correct answer: option 4
It is given that $f(x)$ and $g(x)$ are solutions of the differential equation
$a \frac{d^2 y}{d x^2}+x^2 \frac{d y}{d x}+y=e^x$
∴ $a \frac{d^2}{d x^2}\{f(x)\}+x^2 \frac{d}{d x}\{f(x)\}+f(x)=e^x$
and, $\frac{d^2}{d x^2}\{g(x)\}+x^2 \frac{d}{d x}\{g(x)\}+g(x)= e^x$
$\Rightarrow a \frac{d^2}{d x^2}\{f(x)-g(x)\}+x^2 \frac{d}{d x}\{f(x)-g(x)\}+\{f(x)-g(x)\}=0$
$\Rightarrow f(x)-g(x)$ is a solution of the differential equation
$a \frac{d^2 y}{d x^2}+x^2 \frac{d y}{d x}+y=0$