First slide

Methods of solving first order first degree differential equations

Question

$The solution of the differential equation (e^{x^{2}} + e^{y^{2}}) y \frac{d y}{d x} + e^{x^{2}} (x y^{2} - x) = 0 is$

Moderate

A

$e^{x^{2}} (y^{2} - 1) + e^{y^{2}} = c$
B

$e^{y^{2}} (x^{2} - 1) + e^{x^{2}} = c$
C

$e^{y^{2}} (y^{2} - 1) + e^{x^{2}} = c$
D

$e^{x^{2}} (y - 1) + e^{y^{2}} = c$

Solution

$\begin{array}{l} Put y^{2} = t \\ \Rightarrow (e^{x^{2}} e^{y}) \frac{d t}{d x} + 2 e^{x^{2}} (x t - x) = 0 \\ e^{x^{2}} + e^{t} = 2 e^{x^{2}} x (t - 1) \frac{d x}{d t} = 0 \\ Put e^{x^{2}} = z \\ \Rightarrow e^{x^{2}} 2 x \frac{d x}{d y} = \frac{d z}{d t} \\ ∴ z + e^{t} + \frac{d z}{d t} (t - 1) = 0 \\ \frac{d z}{d t} + \frac{z}{t - 1} = \frac{- e^{t}}{t - 1} \\ I. F = e^{\int \frac{d t}{t - 1}} = t - 1 \\ G.S is e^{x^{2}} (y^{2} - 1) + e^{y^{2}} = c \end{array}$

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