First slide

Methods of solving first order first degree differential equations

Question

$The solution of D.E is \frac{x + y \frac{d y}{d x}}{y - x \frac{d y}{d x}} = \frac{x \cos^{2} (x^{2} + y^{2})}{y^{3}} is$

Difficult

A

$\tan (x^{2} {+y}^{2}) = \frac{x^{2}}{y^{2}} +C$
B

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

$\tan (x^{2} {+y}^{2}) = y^{2} {/x}^{2}$
D

$\cot (x^{2} {+y}^{2}) = y^{2} {/x}^{2} + C$

Solution

$\begin{array}{l} \frac{x + y \frac{d y}{d x}}{y - x \frac{d y}{d x}} = \frac{x}{y^{2} y} \cos^{2} (x^{2} + y^{2}) \\ \frac{x + y \frac{d y}{d x}}{\cos^{2} (x^{2} + y^{2})} = \frac{x}{y} \frac{y - x \frac{d y}{d x}}{y^{2}} \\ \frac{1}{2} \frac{2 x + 2 y \frac{d y}{d x}}{\cos^{2} (x^{2} + y^{2})} = \frac{x}{y} \frac{y - x \frac{d y}{d x}}{y^{2}} \\ \to \frac{1}{2} \int \sec^{2} (x^{2} + y^{2}) d (x^{2} + y^{2}) = \int \frac{x}{y} d (x / y) \\ \to \frac{1}{2} \tan (x^{2} {+y}^{2}) = \frac{\frac{x^{2}}{y^{2}}}{2} +C \end{array}$
$\tan (x^{2} {+y}^{2}) = \frac{x^{2}}{y^{2}} +C$

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