Q120 / 160 · AP EAMCET 2024 (23 May Shift 1) · Mathematics
The solution of the differential equation $e^x y d x+e^x d y+x d x=0$ is
- A$e^x+y x^2=c$
- Correct answerB$2 y e^x+x^2=c$
- C$y e^x+x^2 e^y=c$
- D$e^x+x e^y=c$
Solution
$e^x y d x+e^x d y+x d x=0$
$\Rightarrow\left(e^x y+x\right) d x+e^x d y=0 \Rightarrow \frac{d y}{d x}+y=-x e^{-x}$
I.F. $=e^{\int d x}=e^x$
Solution : $\begin{aligned} & y e^x=\int-x d x+C_1 \Rightarrow y e^x=\frac{-x^2}{2}+C_1 \\ & 2 y e^x+x^2=C \end{aligned}$
Solution : $\begin{aligned} & y e^x=\int-x d x+C_1 \Rightarrow y e^x=\frac{-x^2}{2}+C_1 \\ & 2 y e^x+x^2=C \end{aligned}$
Hence, the correct answer is option: 'B'.