First slide

Types of solutions of differential equations

Question

$The solution of the differential equation (y + x \sqrt{x y} (x + y)) d x + (y \sqrt{x y} (x + y) - x) d y = 0 is$

Moderate

A

$\frac{x^{2} y^{2}}{\sqrt{2}} + \tan^{- 1} \frac{x}{y} = c$
B

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

$\frac{x^{2} + y^{2}}{2} + 2 \tan^{- 1} \sqrt{\frac{x}{y}} = c$
D

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

Solution

$\begin{array}{l} y d x - x d y + x \sqrt{x y} (x + y) d x + y \sqrt{x y} (x + y) d y = 0 \\ y d x - x d y + (x + y) \sqrt{x y} (x d x + y d x) = 0 \\ \frac{y d x - x d y}{y^{2}} + (\frac{x}{y} + 1) \sqrt{\frac{x}{y}} (d (\frac{x^{2} + y^{2}}{2})) = 0 \\ d (\frac{x^{2} + y^{2}}{2}) + \frac{d (\frac{x}{y})}{(\frac{x}{y} + 1) \sqrt{\frac{x}{y}}} = 0 \\ \Rightarrow \frac{x^{2} + y^{2}}{2} + 2 \tan^{- 1} \sqrt{\frac{x}{y}} = c \end{array}$

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