First slide

Methods of solving first order first degree differential equations

Question

$The equation of the curve whose slope is \frac{y - 1}{x^{2} + x} and which passes through the point (1,0) is$

Moderate

A

$xy+x+y-1=0$
B

$xy-x-y-1=0$
C

$(y-1) (x+1) =2x$
D

$y (x+1) -x+1=0$

Solution

$\begin{array}{l} \int \frac{1}{x^{2} + x} dx = \int \frac{dy}{y - 1} \\ \Rightarrow \int \frac{1}{x (x + 1)} dx = \int \frac{dy}{y - 1} \\ \Rightarrow \log x - \log (x + 1) = \log (y - 1) + \log c \\ \Rightarrow \frac{x}{x + 1} = c (y - 1) pass (1, 0) \\ \Rightarrow c = - 1 / 2 \\ \Rightarrow \frac{x}{x + 1} = - 1 / 2 (y - 1) \Rightarrow xy + x + y - 1 = 0 \end{array}$

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