The solution $x^2\frac{dy}{dx}=x^2+xy+y^

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Target Exam

CUET

Subject

-- Mathematics - Section B1

Chapter

Differential Equations

Question:

The solution $x^2\frac{dy}{dx}=x^2+xy+y^2$ is:

Options:

$\tan^{-1}\frac{y}{x}=\log y + c$

$\tan^{-1}\frac{x}{y}=\log x + c$

$\tan^{-1}\frac{x}{y}=\log y + c$

$\tan^{-1}\frac{y}{x}=\log x + c$

Correct Answer:

$\tan^{-1}\frac{y}{x}=\log x + c$

Explanation:

$\frac{dy}{dx}=1+\frac{y}{x}+(\frac{y}{x})^2$ Substitute $\frac{y}{x}=t⇒\frac{dy}{dx}=t+x\frac{dt}{dx}$

$⇒x.\frac{dt}{dx}=1+t^2⇒\int\frac{dt}{1+t^2}=\int\frac{dx}{x}$

$⇒\tan^{-1}t=ln\,x+c⇒\tan^{-1}\frac{y}{x}=ln\,x+c$