First slide

Methods of solving first order first degree differential equations

Question

$For the primitive integral equation ydx + y^{2} dy = xdy, x \in R, y > 0 y = y (x), y (1) = 1 then y (- 3) is$

Moderate

A

3
B

2
C

1
D

5

Solution

$\begin{array}{l} \frac{y d x - x d y}{y^{2}} = - d y \\ \Rightarrow \int d (\frac{x}{y}) = - \int d y \\ \Rightarrow \frac{x}{y} = - y + c \\ \Rightarrow y (1) = 1 \Rightarrow 2 = c \\ \Rightarrow \frac{x}{y} = - y + 2 \\ \Rightarrow y^{2} - 2 y + x = 0 \\ Put x = - 3 \Rightarrow y^{2} - 2 y - 3 = 0 \Rightarrow y = 3, - 1 \end{array}$

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