Observe the following statements I. If $d y+2 x y d x=2 e^{-x^2} d x$, then $y e^{x^2}=2 x+c$ II. If $y…
Observe the following statements
I. If $d y+2 x y d x=2 e^{-x^2} d x$, then $y e^{x^2}=2 x+c$
II. If $y e^{-x^2}-2 x=c$, then $d x=\left(2 e^{-x^2}-2 x y\right) d y$ which of the following is a correct statement?
Both I and II are true
Neither I nor II is true
I is false, but II is true
I is false, II is true
Solution
I. $d y+2 x y d x=2 e^{-x^2} d x$
$\Rightarrow \quad \frac{d y}{d x}+2 x y=2 e^{-x^2}$
This is a linear differential equation in $y$
Here, $P=2 x, Q=2 e^{-x^2}$
$\therefore \text { I.F. }=e^{\int P d x}=e^{\int 2 x d x}=e^{x^2}$
$\therefore$ Complete solution is
$\begin{aligned}
& y e^{x^2}=2 \int e^{-x^2} e^{x^2} d x+c \\
\Rightarrow \quad y e^{x^2} & =2 x+c
\end{aligned}$
II. $y e^{x^2}-2 x=c$
On differentiating w.r.t $x$, we get
$y e^{x^2} \cdot 2 x+e^{x^2} \frac{d y}{d x}-2=0$
$\begin{aligned}
\Rightarrow & e^{x^2} \frac{d y}{d x} & =2-2 x y e^{x^2} \\
\Rightarrow & \frac{d y}{d x} & =2 e^{-x^2}-2 x y
\end{aligned}$
$\therefore \mathrm{I}$ is true and II is false.