Solve the differential equation dy/dx = x^2 + y^2.

Solve the differential equation dy/dx = x^2 + y^2.

Step 1: Identify the type of differential equation. Here, dy/dx = x^2 + y^2 is a non-linear equation.
Step 2: Consider a substitution method. A common substitution for equations like this is to let y = vx, where v is a function of x.
Step 3: Differentiate y = vx with respect to x to find dy/dx. This gives dy/dx = v + x(dv/dx).
Step 4: Substitute dy/dx into the original equation: v + x(dv/dx) = x^2 + (vx)^2.
Step 5: Simplify the equation. This will lead to a new equation in terms of v and x.
Step 6: Solve the new equation for v. This may involve separating variables or using another method.
Step 7: Once you find v, substitute back to find y in terms of x.

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