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  • Given that frac{dy}{dx}=e^{-2y} and y = 0 when x = 5. Find the value of x when y = 3.

Given that \frac{dy}{dx}=e^{-2y} and y = 0 when x = 5.
Find the value of x when y = 3.

Answers (1)

Given:

 \frac{dy}{dx}=e^{-2y}

(5,0) is a solution of this equation

To find: Solution of the given differential equation

Rewriting the equation.

\frac{dy}{e^{-2y}}=dx

Integrate on both the sides,

\int \frac{dy}{e^{-2y}}=\int dx\\ \Rightarrow \int e^{2y}dy=\int dx

Formula:

 \int e^{ax} dx =\frac{1}{a}e^{ax}\\ \frac{e^{2y}}{2}=x+c

Given (5,0) is a solution so to get c, satisfying these values

\frac{1}{2}=5+c\\ c=-\frac{9}{2}\\

Hence the solution is

e2y=2x + 9

when y=3,

e2(3) =2x + 9

e6=2x + 9

e6+ 9=2x

\Rightarrow x=\frac{e^{6}+9}{2}

 

Posted by

infoexpert24

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