First slide

Methods of solving first order first degree differential equations

Question

The solution of $y (2 x^{2} y + e^{x}) d x - (e^{x} + y^{3}) d y = 0$ , if $y (0) = 1$ is

Moderate

A

$6 e^{x} + 4 x^{3} y - 3 y^{3} - 3 y = 0$
B

$y^{2} e^{x} - 4 x y - 3 x^{3} - 3 = 0$
C

$x^{3} e^{x} - 4 x^{3} y - 3 x y^{3} - 3 x = 0$
D

none of these

Solution

$\begin{array}{l} y e^{x} d x - e^{x} d y + 2 x^{2} y^{2} d x - y^{3} d y = 0 \\ \Rightarrow \frac{y e^{x} d x - e^{x} d y}{y^{2}} + 2 x^{2} d x - y d y = 0 \\ d (\frac{e^{x}}{y}) + 2 x^{2} d x - y d y = 0 \\ Integrating we get, \\ \frac{e^{x}}{y} + \frac{2 x^{3}}{3} - \frac{y^{2}}{2} = C \\ y (0) = 1 \Rightarrow 1 + 0 - \frac{1}{2} = C given C = \frac{1}{2} \\ ∴ \frac{e^{x}}{y} + \frac{2 x^{3}}{3} - \frac{y^{2}}{2} = \frac{1}{2} \\ Gives 6 e^{x} + 4 x^{3} y - 3 y^{3} - 3 y = 0 \end{array}$

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