Which one of the following is a linear differential equation?
Which one of the following is a linear differential equation?
$\frac{d x}{d y}+y^2=e^{e^x}$
$d r+\left(2 r^2 \cot \theta+\sin 2 \theta\right) d \theta=0$
$\frac{d y}{d x}=e^{x-y}\left(e^x-e^{-y}\right)$
$x^2 d y+x y d x-1=0$
Solution
The standard form of linear differential equation is
$P \frac{d y}{d x}+Q y=R$ or $P \frac{d x}{d y}+Q x=R$ where $P, Q$ and $R$ are the function of $x$ only.
Option (d) can be written as
$\begin{aligned} & x^2 d y+x y d x-d x=0 \\ & x^2 \frac{d y}{d x}+x y-1=0 \\ & \Rightarrow x^2 \frac{d y}{d x}+x y=1\end{aligned}$
Which follows the standard form of linear differential equation.