First slide

Types of solutions of differential equations

Question

$The solution of differential equation \frac{x + y y^{1}}{y - x y^{1}} = x^{2} + 2 y^{2} + \frac{y^{4}}{x^{2}} is$

Moderate

A

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

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

$\ln (x^{2} + y^{2}) = \frac{2 y}{x} + c$
D

$\ln (x^{2} + y^{2}) = \frac{2 x}{y} = c$

Solution

$\begin{array}{l} \frac{x d x + y d y}{y d x - d x y} = x^{2} + 2 y^{2} + \frac{y^{4}}{x^{2}} \\ = \frac{x^{4} + 2 x^{2} y^{2} + y^{4}}{x^{2}} = \frac{{(x^{2} + y^{2})}^{2}}{x^{2}} \Rightarrow \frac{x d x + y d y}{{(x^{2} + y^{2})}^{2}} = \frac{y d x - x d y}{x^{2}} \\ \frac{1}{2} \int \frac{2 x d x + 2 y d y}{{(x^{2} + y^{2})}^{2}} = \frac{x d y - y d x}{x^{2}} \\ Integrations on both sides \frac{1}{2 (x^{2} + y^{2})} = \frac{y}{x} + c \Rightarrow \frac{1}{x^{2} + y^{2}} = \frac{2 y}{x} + c \end{array}$

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