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JEE MainMathematicsEasyDifferential equation

The general solution of the differential equation
(y2−x3)dx−xydy=0(x≠0)\left( y^{2} - x^{3} \right)dx - xydy = 0(x \neq 0) is:(where c is a
constant of integration)

Show answer and solution
(y2−x3)dx−xydy=0⇒y(ydx−xdy)=x3dx⇒yx(ydx−xdyx2)=dx\left( y^{2} - x^{3} \right)dx - xydy = 0 \Rightarrow y(ydx - xdy) = x^{3}dx \Rightarrow \frac{y}{x}\left( \frac{ydx - xdy}{x^{2}} \right) = dx
⇒−yxd(yx)=dx⇒−12(yx)2=x+k⇒−y2=2x3+2x2k⇒y2+2x3+cx2=0\Rightarrow - \frac{y}{x}d\left( \frac{y}{x} \right) = dx \Rightarrow - \frac{1}{2}\left( \frac{y}{x} \right)^{2} = x + k \Rightarrow - y^{2} = 2x^{3} + 2x^{2}k \Rightarrow y^{2} + 2x^{3} + cx^{2} = 0

Source: Grass-G / M-28 Mathematics Paper FST 16 Jan.docx

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