The solution of differential equation x^{2}ydx-\left ( x^{3}+y^{2} \right )dy=0 is

  • Option 1)

    \frac{1}{3}\frac{x^{3}}{y^{3}}+log \:y=C

  • Option 2)

    \frac{1}{3}\frac{x^{3}}{y^{3}}-log \:y=C

  • Option 3)

    \frac{x^{3}}{y^{3}}+log \:y=-C

  • Option 4)

    None of these

 

Answers (1)

The solution of differential equation x^{2}ydx-\left ( x^{3}+y^{2} \right )dy=0 is

x^{2}ydx-(x^{3}+y^{3})dy=0\\ \therefore \frac{dx}{dy}=\frac{x^{3}+y^{3}}{x^{2y}}\\

Divide by y3

\frac{dx}{dy}=\frac{1+(\frac{x}{y})^{3}}{(\frac{x}{y})^{2}}

Now let \frac{x}{y}=v=x=vy\\ \frac{dx}{dy}=\frac{1+V^{3}}{V^{2}}\\ y\frac{dv}{dy}=\frac{1+V^{3}}{V^{2}}-V=\frac{1+V^{3}-V^{3}}{V^{2}}=\frac{1}{V^{2}}\\

\int v^{2}dv=\int \frac{dy}{y}\\ \therefore \frac{V^{3}}{3}=logy+c\\ \frac{1}{3}.\frac{x^{3}}{y^{3}}-logy=c


Option 1)

\frac{1}{3}\frac{x^{3}}{y^{3}}+log \:y=C

This solution is incorrect 

Option 2)

\frac{1}{3}\frac{x^{3}}{y^{3}}-log \:y=C

This solution is correct 

Option 3)

\frac{x^{3}}{y^{3}}+log \:y=-C

This solutioin is incorrect 

Option 4)

None of these

This solution is incorrect 

 

 

 

 

Preparation Products

Knockout BITSAT 2021

It is an exhaustive preparation module made exclusively for cracking BITSAT..

₹ 4999/- ₹ 2999/-
Buy Now
Knockout BITSAT-JEE Main 2021

An exhaustive E-learning program for the complete preparation of JEE Main and Bitsat.

₹ 27999/- ₹ 16999/-
Buy Now
Exams
Articles
Questions