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Solve the following $\cos^{2}xdy +ye^{\tan x }dx=0 $

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Toolbox:
  • First order , first degree DE
  • Variable separable : Variables of a DE are rearranged to separate then, ie
  • $f_1(x)g_2(y)dx+f_2(x)g_1(y)dy=0$
  • Can be written as $ \large\frac{g_1 (y)}{g_2(y)}$$dy=-\large\frac{f_1(x)}{f_2(x)}$$dx$
  • The solution is therefore $\int \large\frac{g_1(y)}{g_2(y)}$$dy=-\int \large\frac{f_1(x)}{f_2(x)}$$dx+c$
Step 1:
$\cos ^2 x dy+y e^{\tan x}dx=0$ divided by $y \cos ^2 x$
$\large\frac{dy}{y}+\large\frac{e^{\tan x}}{\cos^2 x}$$dx$$=0$
Step 2:
The variables have been seperated
$\int \large\frac{dy}{y} =- \int e^{\tan x} $$\sec^2 x dx +c$
$\log y=-e^{\tan x}+c$
answered Sep 4, 2013 by meena.p
 
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