In each of the following differential equation show that it is homogeneous and solve it. $x d y=(x+y) d x$

Home » In each of the following differential equation show that it is homogeneous and solve it.

In each of the following differential equation show that it is homogeneous and solve it. $x d y=(x+y) d x$

Solution:

$\begin{array}{l} x d y=(x+y) d x \\ \frac{d y}{d x}=\frac{x+y}{x} \\ \Rightarrow \frac{d y}{d x}=1+\frac{y}{x} \\ \Rightarrow \frac{d y}{d x}=f\left(\frac{y}{x}\right) \end{array}$
$\Rightarrow$ the given differential eq. is a homogenous eq.
The solution of the given differential eq. is:
Putting $y=v x$
$\begin{array}{l} \Rightarrow \frac{d y}{d x}=v+x \frac{d v}{d x} \\ \Rightarrow v+x \frac{d v}{d x}=1+\frac{v x}{x} \\ \Rightarrow x \frac{d v}{d x}=1+v-v \\ \Rightarrow x \frac{d v}{d x}=1 \\ \Rightarrow d v=\frac{d x}{x} \end{array}$
On integrating both sides we obtain:
$\begin{array}{l} \int \mathrm{dv}=\int \frac{\mathrm{dx}}{\mathrm{x}}+c \\ \mathrm{v}=\ln |\mathrm{x}|+\mathrm{c} \end{array}$
By resubstituting the value of $y=v x$ we obtain
$\begin{array}{l} \frac{y}{x}=\ln |x|+c \\ y=x \ln |x|+c x \end{array}$