Question

Assertion :The solution of the equation $x\frac{dy}{dx}+6y=3x{y}^{4/3}$ is $y\left(x\right)=\frac{1}{(x+C{x}^2{)}^3}$ Reason: The solution of a linear equation is obtained by multiplying with its integrating factor.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

The given differential equation is not a linear equation. Dividing by $y^{4/3}$, we have
$\frac{x}{y^{4/3}}\frac{dy}{dx}+6y^{-1/3}=3x$
Put $y^{-1/3}=v\Rightarrow y^{-4/3}\frac{dy}{dx}=-3\frac{dv}{dx}$, so
$-3x\frac{dv}{dx}+6v=3x$
$\Rightarrow \frac{dv}{dx}-\frac{2}{x}v=-1$
which is a linear equation whose integrating factor is $x^{-2}$.
$x^{-2}v=\frac{1}{x}+C$
$\Rightarrow v=x+C{x}^2$
$\Rightarrow y\left(x\right)=\frac{1}{(x+C{x}^2{)}^3}$