First slide

Methods of solving first order first degree differential equations

Question

$The solution of the differential equation \frac{d y}{d x} = x y + x + y + 1 is$

Moderate

A

$y = \frac{1}{2} x^{2} + x + C$
B

$\log (y + 1) = \frac{1}{2} x^{2} + x + C$
C

$C (y + 1) = e^{x^{2} + \frac{2 x}{2}}$
D

$y = \frac{1}{2} x^{3} + x + C$

Solution

$\frac{d y}{d x} = x y + x + y + 1$
$\begin{array}{l} = x (y + 1) + (y + 1) \\ = (x + 1) (y + 1) \end{array}$
$\Rightarrow \frac{d y}{y + 1} = (x + 1) d x$
Integration on both sides
$\Rightarrow \int \frac{d y}{y + 1} = \int d x$
$log (y + 1) = \frac{x^{2}}{2} + x + C$

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