Question

Solutions of the differential equation $x^2{\left(\frac{dy}{dx}\right)}^2+xy\left(\frac{dy}{dx}\right)-6y^2=0$ are:

• A. $y=c{x}^2$
• B. $x^{3}y=c$
• C. $x{y}^3=c$
• D. $y=cx$

Answer 1

Answer: (A), (B)

The correct options are
A $y=c{x}^2$
B $x^{3}y=c$

$x^2{\left(\frac{dy}{dx}\right)}^2-6y^2+xy\frac{dy}{dx}=0$
$\Rightarrow \frac{dy}{dx}=-\frac{3y}{x}$ or $\frac{dy}{dx}=\frac{2y}{x}$
For $\frac{dy}{dx}=-\frac{3y}{x}\Rightarrow \int \frac{dy}{dx}\frac{1}{y}dx=\int -\frac{3}{x}dx$
$\Rightarrow \log y=-3\log x+c\Rightarrow y=\frac{c}{x^3}$
and for $\frac{dy}{dx}=\frac{2y}{x}\Rightarrow \int \frac{dy}{dx}\frac{1}{y}dx=\int \frac{1}{x}dx$
$\Rightarrow \log y=2\log x+c\Rightarrow y=c{x}^2$