Q77 / 180 · KCET 2020 · Mathematics
The general solution of the differential equation $x^{2} d y-2 x y d x=x^{4} \cos x d x$ is
- Correct answerA$y=x^{2} \sin x+c x^{2}$
- B$y=x^{2} \sin x+c$
- C$y=\sin x+c x^{2}$
- D$y=\cos x+c x^{2}$
Solution
We have,
$\begin{aligned}
x^{2} d y-2 x y d x &=x^{4} \cos x d x \\
\Rightarrow \quad \quad \frac{d y}{d x}-\frac{2}{x} y &=x^{2} \cos x
\end{aligned}$
Comparing with $\frac{d y}{d x}+P y=Q$
Where $P=-\frac{2}{x}$ and $Q=x^{2} \cos x$
$\therefore \quad \text { IF }=e^{\int \frac{-2}{x} d x}=e^{-2 \log x}=e^{\log \left(\frac{1}{x^{2}}\right)}=\frac{1}{x^{2}}$
$\therefore$ The solution of the given differential
$y \cdot \operatorname{IF}=\int(Q \times \operatorname{IF}) d x+c$
$\Rightarrow y \times \frac{1}{x^{2}}=\int\left(x^{2} \cos x\right)\left(\frac{1}{x^{2}}\right) d x+c$
$\Rightarrow \quad \frac{y}{x^{2}}=\int \cos x+c$
$\Rightarrow \quad \frac{y}{x^{2}}=\sin x+c$
$\Rightarrow \quad y=x^{2} \sin x+c x^{2}$
Hence, the correct answer is option: 'A'.