# 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...

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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### 1 Answer

`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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