First slide

Types of solutions of differential equations

Question

$The solution of differential equation \frac{x d y - y d x}{x^{2} + y^{2}} + {(\tan^{- 1} (\frac{x}{y}))}^{3} (x d y + y d x) = 0 is$

Moderate

A

$\tan^{- 1} \frac{y}{x} = k e^{- x y}$
B

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

$\tan^{- 1} \frac{x}{y} = k e^{x y}$
D

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

Solution

$\begin{array}{l} \frac{d (\tan^{- 1} \frac{y}{x})}{{(\tan^{- 1} \frac{y}{x})}^{3}} + d (x y) = 0 \\ Integrating both sides \frac{- 1}{2 {(\tan^{- 1} \frac{y}{x})}^{2}} + x y = c \end{array}$

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