## To find the value of y when x=2 for the equation dy/dx = 2y^2, you first integrate both sides of the equation with respect to x.

Integrating dy/dx gives you âˆ«dy = âˆ«2y^2 dx. This simplifies to y = âˆ«2y^2 dx.

Now, integrating the right side requires knowledge of the integral of y^2. To integrate y^2, use the power rule for integration: âˆ«y^n dy = (1/(n+1)) * y^(n+1). Applying this to the integral of 2y^2 gives (2/(2+1)) * y^(2+1) = (2/3) * y^3 + C, where C is the constant of integration.

To complete the solution, you need to find the value of C. Use the given information that y=-1 when x=1. Substituting these values into the equation y = (2/3) * y^3 + C gives -1 = (2/3) * (-1)^3 + C.

Solving this equation for C, you get C = -1 - (2/3). Simplifying, C = -5/3.

Now that you have the value of C, you can find the value of y when x=2. Substituting x=2 and the value of C into the equation y = (2/3) * y^3 + C gives y = (2/3) * y^3 - 5/3. To solve for y, you can use numerical techniques or approximation methods as the equation is non-linear.