First slide

Methods of solving first order first degree differential equations

Question

$The solution of the differential equation \frac{d y}{d x} = \frac{(1 + y)^{2}}{x (1 + y) - x^{2}} is given as \ln (1 + y) =$

Moderate

A

$\frac{1 + y}{x} + c$
B

$\frac{1 - y}{x} + c$
C

$\frac{- 1 - y}{x} + c$
D

None of these

Solution

$\begin{array}{l} consider \frac{d x}{d y} = \frac{x}{1 + y} - \frac{x^{2}}{(1 + y)^{2}} \\ \Rightarrow - \frac{1}{x^{2}} \frac{d y}{d x} + \frac{1}{x (1 + y)} = \frac{1}{(1 + y)^{2}} \\ Substituting \frac{1}{x} = t, we get \frac{d t}{d y} + \frac{t}{1 + y} = \frac{1}{(1 + y)^{2}}; \end{array}$
$Which is linear I.F = e^{\int \frac{1}{1 + y} d y} = (1 + y)$
$\Rightarrow solution of differential equation is$
$t (1 + y) = \int \frac{1}{1 + y} d y = c \Rightarrow c + \frac{1 + y}{x} = \ln (1 + y)$

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