This is a separable first order differential equation as an initial value problem.
[tex] \dfrac{dy}{dx} =2xy^2[/tex]
Divide both sides by y².
[tex]\dfrac{1}{y^2}\dfrac{dy}{dx}=2x[/tex]
Multiply both sides by dx.
[tex]\dfrac{1}{y^2} dy=2x \times dx[/tex]
Integrate both sides.
[tex]\int \dfrac{1}{y^2} dy= \int 2x \times dx[/tex]
[tex]- \dfrac{1}{y}=x^2+C[/tex]
Multiply both sides by -1
[tex]\dfrac{1}{y}=-x^2+C_1[/tex]
(where [tex]C_1=-C[/tex])
Now, isolate y on one side of the equation.
[tex]y=\dfrac{1}{-x^2+C_1}[/tex]
That's the general solution. Now, plug in the value x=-1 and y=2.
[tex]2=\dfrac{1}{C_1-1}[/tex]
[tex]C_1=3/2[/tex]
The final solution is the following:
[tex]y=\dfrac{1}{-x^2+3/2}[/tex]
I hope this helps! If you need any clarifications, feel free to comment! :)
