Question

The general solution of the differential equation $ydx-xdy+\ln xdx=0$ is
(where $C$ is constant of integration)

• A. $y+\ln x-1=C$
• B. $y+\ln x-1=Cx$
• C. $y+\ln x+1=Cx$
• D. $y-\ln x-1=C$

Answer 1

The correct option is C $y+\ln x+1=Cx$

Given differential equation
$ydx-xdy+\ln xdx=0$
Dividing by $x^2,$ we get
$\Rightarrow \frac{xdy-ydx}{x^2}-\frac{\ln x}{x^2}dx=0$
$\Rightarrow d\left(\frac{y}{x}\right)-\frac{\ln x}{x^2}dx=0$
Integrating both sides, we have
$\frac{y}{x}-\int \frac{\ln x}{x^2}dx=0$
$\Rightarrow \frac{y}{x}+\frac{1}{x}\ln x-\int \frac{1}{x^2}dx=0$
$\Rightarrow \frac{y}{x}+\frac{1}{x}\ln x+\frac{1}{x}=C$
$\Rightarrow y+\ln x+1=Cx$