Solve the differential equation ( e^ - 2 x x - y x ) dx dy = 1 (x… - Vidyadip

Question

Solve the differential equation $\left( {\frac{{{e^{ - 2\sqrt x }}}}{{\sqrt x }} - \frac{y}{{\sqrt x }}} \right)\frac{{dx}}{{dy}}$ = 1 (x $\neq$ 0)

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Answer

$\left( {\frac{{{e^{ - 2\sqrt x }}}}{{\sqrt x }} - \frac{y}{{\sqrt x }}} \right)\frac{{dx}}{{dy}}$ = 1
$\frac{{dy}}{{dx}} = \frac{{{e^{ - 2\sqrt x }}}}{{\sqrt x }} - \frac{y}{{\sqrt x }}$
$\frac{{dy}}{{dx}} + \frac{y}{{\sqrt x }} = \frac{{{e^{ - 2\sqrt x }}}}{{\sqrt x }}$
Given differential equation is of the form:
$\frac{{dy}}{{dx}} + Py = Q$
$I.F = {e^{\int {\frac{1}{{\sqrt x }}dx} }} = {e^{2\sqrt x }}$
Solution is,
$y \times {e^{2\sqrt x }} = \int {{e^{2\sqrt x }} \times \frac{{{e^{ - 2\sqrt x }}}}{{\sqrt x }}dx + c}$
$y{e^{2\sqrt x }} = \int {\frac{1}{{\sqrt x }}dx + C} $
$y{e^{2\sqrt x }} = 2\sqrt x + C$

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