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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The differential equation `dy/dx= 8x^2y^2` has to be solved.

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

=> `(1/y^2)*dy = 8*x^2*dx`

take the integral of both the sides

`int (1/y^2)*dy = 8*int x^2*dx`

=> `-1/y = 8*x^3/3`

=> `y = -3/(8x^3) + C`

It is given that y(2) = 3

`3 = -3/64 + C`

=> `C = 3 + 3/64 = 195/64`

**The function `y = -3/(8x^3) + 195/64` **

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

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

`-1/y=8/3 x^3`

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

Since: `y(2)=3 rArr` `3=-3/64 +c`

`c=189/64`

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