First slide

Types of solutions of differential equations

Question

$Solution of differential equation \frac{x^{2}}{(\frac{1}{x^{2}} + y^{2})} (x d y + y d x) + y^{2} (x d y - y d x) = 0 is$

Moderate

A

$\tan^{- 1} x y - \frac{1}{2} {(\frac{y}{x})}^{2} = c$
B

$\tan^{- 1} x y + \frac{1}{2} {(\frac{y}{x})}^{2} = c$
C

$\tan^{- 1} x y + \frac{1}{3} {(\frac{y}{x})}^{3} = c$
D

$None of these$

Solution

$\begin{array}{l} Given differential equation is \\ \frac{x^{2}}{(\frac{1}{x^{2}} + y^{2})} (x d y + y d x) + y^{2} (x d y - y d x) = 0 \\ \Rightarrow \frac{x^{4}}{(1 + x^{2} y^{2})} (x d y + y d x) + y^{2} (x d y - y d x) = 0 \\ \Rightarrow \frac{1}{1 + x^{2} y^{2}} (x d y + y d x) + \frac{y^{2}}{x^{2}} (\frac{x d y - y d x}{x^{2}}) = 0 \\ \Rightarrow d (\tan^{- 1} x y) + \frac{1}{3} d {(\frac{y}{x})}^{3} = 0 \\ \Rightarrow \tan^{- 1} (x y) + \frac{1}{3} {(\frac{y}{x})}^{3} = c \end{array}$

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