Solve the following differential equations: (1-x^2)dy + xy dx = x… - Vidyadip

Question

Solve the following differential equations:

$(1-\text{x}^2)\text{dy + xy dx = xy}^2\text{ dx}$

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Answer

$(1-\text{x}^2)\text{dy + xy dx = xy}^2\text{dx}$
$(1-\text{x}^2)\text{dy = dx}(\text{xy}^2-\text{xy})$
$(1-\text{x}^2)\text{dy = xy(y}-1)\text{dx}$
$\int\frac{\text{dy}}{\text{y(y}-1)}=\int\frac{\text{xdx}}{1-\text{x}^2}$
$\int\Big(\frac{1}{\text{y}-1}-\frac{1}{\text{y}}\Big)\text{dy}=\frac{1}{2}\int\frac{2\text{x}}{1-\text{x}^2}\text{dx}$
$\int\Big(\frac{1}{\text{y}-1}-\frac{1}{\text{y}}\Big)\text{dy}=-\frac{1}{2}\int\frac{-2\text{x}}{1-\text{x}^2}\text{dx}$
$\log|\text{y}-1|-\log|\text{y}|=-\frac{1}{2}\log|1-\text{x}^2|+\text{C}$

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