Question

Let $y=f\left(x\right)$ be a function satisfying the differential equation $\frac{xdy}{dx}+2y=4x^2$ and $f\left(1\right)=1$, then $f(-3)$ is equal to.

• A. $-3$
• B. $0$
• C. $3$
• D. $9$

Answer 1

The correct option is D $9$

$\frac{dy}{dx}+\left(\frac{2}{x}\right)y=4x$
(Linear differentiable equation)
I.F. $=e^{2\int \frac{dx}{x}}=x^2$
So, $y\cdot (x{)}^2=\int \left(4x\right)x^{2}dx+c$
$\Rightarrow y\left(x^2\right)=x^4+x$
As, $y\left(1\right)=1\Rightarrow 1=1+c\Rightarrow c=0$
So, $f\left(x\right)=x^2\Rightarrow f(-3)=9$.