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

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embizze eNotes educator| Certified Educator

Solve `(dy)/(dx)=8x^2y^2` with the initial condition y(2)=3:

This is separable so we get all expressions involving y on one side of the equation, and x on the other:

`(dy)/(y^2)=8x^2dx`    Integrate both sides:

`int y^(-2)dy=8int x^2dx`


From the initial condition, when x=2 y=3 so:



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

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



We can check this by differentiation:

Note that `y^2=9/((8x^3-65)^2)`




`=8x^2y^2` as required.