Question

The general solution of the differential equation $(y^2–x^3)dx–xydy=0(x\ne 0)$ is : (where $c$ is a constant of integration)

• A. $y^2+2x^2+c{x}^3=0$
• B. $y^2-2x^3+c{x}^2=0$
• C. $y^2+2x^3+c{x}^2=0$
• D. $y^2–2x^2+c{x}^3=0$

Answer 1

The correct option is C $y^2+2x^3+c{x}^2=0$

$(y^2–x^3)dx–xydy=0\Rightarrow y(ydx-xdy)=x^{3}dx\Rightarrow \frac{y}{x}\left(\frac{ydx-xdy}{x^2}\right)=dx\Rightarrow -\frac{y}{x}d\left(\frac{y}{x}\right)=dx\Rightarrow -\frac{1}{2}{\left(\frac{y}{x}\right)}^2=x+C\Rightarrow y^2+2x^3+c{x}^2=0,\text{where}c=2C$