First slide

Types of solutions of differential equations

Question

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

Moderate

A

$x^{2} y^{2} + \frac{x^{3} y^{3}}{3} + c$
B

$x y^{2} + \frac{x^{3} y^{3}}{3} + c$
C

$x^{2} y + \frac{x^{3} y^{3}}{3} = c$
D

$x^{2} y^{2} + \frac{x y}{3} + c$

Solution

$\begin{array}{l} 2 y d x + x d y + x y^{3} d x + x^{2} y^{2} d y = 0 multiplying by x \\ 2 x y d x + x^{2} d y + x^{2} y^{2} (y d x + x d y) = 0 \\ d (x^{2} y) + x^{2} y^{2} d (x y) = 0 x^{2} y + \frac{x^{3} y^{3}}{3} = c \end{array}$

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