Calculus II · Introduction to differential equations
Stop the separable solution at its first blow-up
Problem
Solve the initial-value problem \[ y'=x(1+y^2),\qquad y(0)=0. \] Derive an explicit formula, verify it by differentiation and the initial condition, and find the maximal open interval containing $0$ on which this IVP solution is finite and differentiable. Explain why the antiderivative equation does not justify calling the resulting tangent formula a global real-valued solution.
Hint
Separate as $dy/(1+y^2)=x\,dx$.
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