Question

If dy/dx=2xy^2 and y(-1)=2, find y(2)

Answer 1

This is a separable first order differential equation as an initial value problem.

$\dfrac{dy}{dx} =2xy^2$

Divide both sides by y².

$\dfrac{1}{y^2}\dfrac{dy}{dx}=2x$

Multiply both sides by dx.

$\dfrac{1}{y^2} dy=2x \times dx$

Integrate both sides.

$\int \dfrac{1}{y^2} dy= \int 2x \times dx$

$- \dfrac{1}{y}=x^2+C$

Multiply both sides by -1

$\dfrac{1}{y}=-x^2+C_1$
(where $C_1=-C$)

Now, isolate y on one side of the equation.

$y=\dfrac{1}{-x^2+C_1}$

That's the general solution. Now, plug in the value x=-1 and y=2.

$2=\dfrac{1}{C_1-1}$

$C_1=3/2$

The final solution is the following:

$y=\dfrac{1}{-x^2+3/2}$

I hope this helps! If you need any clarifications, feel free to comment! :)