First slide

Types of solutions of differential equations

Question

$The solution of d.E y (2 x^{2} y + e^{x}) d x - (e^{x} + y^{3}) d y = 0 is$

Moderate

A

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

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

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

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

Solution

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

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