Question

solve the differential equations
dy/dx=y tan 2x , y (0) = 2

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Answer 1

Answer:

$\sf y^2=4\times sec\:2x$

Step-by-step explanation:

Given:

$\sf \dfrac{dy}{dx} =y\:tan\:2x$

To Find:

The solution of the differential equation when y (0) = 2

Solution:

Solving the equation by making it into a variable separable form,

$\sf \dfrac{dy}{y} =tan\:2x\:dx$

Integrating on both sides we get,

$\displaystyle \sf \int\limits {\dfrac{dy}{y} } \, =\int\limits {tan\:2x} \, dx$

We know that,

$\displaystyle \sf \int\limits {\dfrac{1}{x} } \, dx =log|x|+C$

$\displaystyle \sf \int\limits {tan\:x} \, dx =log|sec\:x|+C$

Hence,

$\sf log|y|=\dfrac{log|sec\:2x|}{2} +C$

$\sf log|y|-\dfrac{1}{2}\times log|sec\:2x|=log\:C$

We know that,

$\sf log\:a-log\:b=log\bigg(\dfrac{a}{b} \bigg)$

$\sf \dfrac{1}{2}\times log\:a=log\sqrt{a}$

Hence,

$\sf log\bigg(\dfrac{y}{\sqrt{sec\:2x} } \bigg)=log\:C$

$\sf \dfrac{y}{\sqrt{sec\:2x} } =\:C$

Given that when x = 0, y = 2. Therefore,

$\sf \dfrac{2}{\sqrt{sec\:2\times 0} } =\:C$

$\sf C=2$

Hence,

$\sf \dfrac{y}{\sqrt{sec\:2x} } =2$

$\sf y=2\times \sqrt{sec\:2x}$

Squaring on both sides,

$\sf y^2=4\times sec\:2x$

Hence the solution of the given differential equation is y² = 4 sec 2x