Solve the following differential equations: xy dy dx =1+x + y + x… - Vidyadip

Question

Solve the following differential equations:
$\text{xy}\frac{\text{dy}}{\text{dx}}=1+\text{x + y + xy}$

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Answer

$\text{xy}\frac{\text{dy}}{\text{dx}}=1+\text{x + y + xy}$
$=(1+\text{x})+\text{y}(1+\text{x})$
$\text{xy}\frac{\text{dy}}{\text{dx}}=(1+\text{x})(1+\text{y})$
$\int\frac{\text{ydy}}{\text{y}+1}=\int\frac{1+\text{x}}{\text{x}}\text{dx}$
$\int\Big(1-\frac{1}{\text{y}+1}\Big)\text{dy}=\int\Big(\frac{1}{\text{x}}+1\Big)\text{dx}$
$\text{y}-\log|\text{y}+1|=\log|\text{x}|+\text{x}+\log|\text{c}|$
$\text{y}=\log|\text{cx(y+1})|+\text{x}$

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