First slide

Methods of solving first order first degree differential equations

Question

$Solve (x y^{2} - e^{\frac{1}{x^{3}}}) d x - x^{2} y d y = 0$

Moderate

A

$3 x^{2} = 2 y^{2} e^{1 / x^{2}} + c x^{2}$
B

$3 y^{2} = 2 x^{2} e^{1 / x^{3}} + c x^{2}$
C

$y = 2 x^{2} e^{1 / x^{3}} + c x^{2}$
D

$x^{2} = 2 y e^{1 / x^{2}} + c x^{2}$

Solution

$\begin{array}{l} Differential Equation can be written as y \frac{d y}{d x} - \frac{y^{2}}{x} = \frac{- e^{1 / x^{3}}}{x^{2}} \\ Put y^{2} = t \Rightarrow y \frac{d y}{d x} = \frac{1}{2} \frac{d t}{d x} \\ I.F = e^{- 2 \int 1 / x d x} = \frac{1}{x^{2}} \\ t \frac{1}{x^{2}} = \int \frac{1}{x^{2}} \cdot \frac{- 2}{x^{2}} e^{1 / x^{3}} d x = \int \frac{- 2}{x^{4}} e^{\frac{1}{x^{3}} d x} \\ \frac{y^{2}}{x^{2}} = \frac{2}{3} e^{1 / x^{3}} + c \Rightarrow 3 y^{2} = 2 x^{2} e^{\frac{1}{x^{3}}} + c x^{2} \end{array}$

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