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1. Find in implicit form the general solution of the differential equation   dy/dx = -...

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user1548464 | (Level 2) eNoter

Posted March 24, 2013 at 1:08 PM via web

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1. Find in implicit form the general solution of the differential equation

 

dy/dx = - 4e^-2y(1+e^4x)/(4x+e^4x)^3

2. Find the corresponding particular solution (in implicit form) that satisfies the initial condition y=0 when x=0

3. Find the explicit form of this particular solution

4. What is the value of y given by this particular solution when x=0.4

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pramodpandey | College Teacher | (Level 3) Valedictorian

Posted March 24, 2013 at 1:34 PM (Answer #1)

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`dy/dx=(-4e^(-2y)(1+e^(4x)))/(4x+e^(4x))^3dx`  ,let us write this equation as variables are separable.

`dy/e^(-2y)=(-4(1+e^(4x)))/(4x+e^(4x))^3dx`

`e^(2y)dy=(-4(1+e^(4x)))/(4x+e^(4x))^3dx`  ,integrating both side with resp. to y and x .

`inte^(2y)dy=int(-4(1+e^(4x)))/(4x+e^(4x))^3dx`  

`(1/2)e^(2y)=-(4x+e^(4x))^(-3+1)/(-2)+c`

`e^(2y)=(4x+e^(4x))^(-2)+c`  ,it is given that y=0 when x=0

`e^(2.0)=(4.0+e^(4.0))^(-2)+c`

`1=1^(-2)+c,`

`1=1+c`

`c=0`  ,so implicit solution is

`e^(2y)=(4x+e^(4x))^(-2)`

`e^(2y)(4x+e^(4x))^(2)=1 `

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