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Solve the differential equation dy/dx=8x^2y^2 with the condition that y(2)=3 the...

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Solve the differential equation dy/dx=8x^2y^2 with the condition that y(2)=3 the solution to the equation is y=

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We have given

`(dy)/(dx)=8x^2y^2`  ,y(2)=3

In given equation ,variables are separable.

`(dy)/y^2=8x^2dx`

Integrate both side ,we have

`int(1/y^2)dy=int8x^2dx+c`

`y^(-2+1)/(-1)=8x^(2+1)/3+c`

`(-1)/y=(8/3)x^3+c`

given when x=2,y=3 ,so we have

`(-1)/3=(8/3)8+c`

`(-65)/3=c`

`(-1)/y=(8/3)x^3-65/3`

`y=(-3)/(8x^3-65)`

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