First slide

Methods of solving first order first degree differential equations

Question

$The solution of the D.E 2 x^{2} y \frac{dy}{dx} = \tan (x^{2} y^{2}) - 2 {xy}^{2}, given that y (1) = \sqrt{\frac{π}{2}} is$

Difficult

A

${sin(x}^{2} y^{2} {)=e}^{x+1}$
B

${sin(x}^{2} y^{2})=x$
C

${cos(x}^{2} y^{2})+x=0$
D

${sin(x}^{2} y^{2} {)=e}^{x-1}$

Solution

$w e k n o w \frac{d}{d x} x^{2} y^{2} = x^{2} 2 y \frac{d y}{d x} + y^{2} 2 x$
$G i v e n e q u a t i o n 2 x^{2} y \frac{d y}{d x} + 2 x y^{2} = \tan (x^{2} y^{2})$
$\begin{array}{l} \frac{d}{d x} (x^{2} y^{2}) = \tan (x^{2} y^{2}) \\ \Rightarrow \int \cot (x^{2} y^{2}) d (x^{2} y^{2}) = \int d x \\ \Rightarrow \log \sin (x^{2} y^{2}) = x + C \\ Put x = 1, y = \sqrt{\frac{π}{2}} \\ \log \sin \frac{π}{2} = 1 + C \\ 0=1+C then C = - 1 \\ \log \sin (x^{2} y^{2}) = x - 1 \\ \sin (x^{2} y^{2}) = e^{x - 1} \end{array}$

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