A function $y$ satisfies the differential equation $y'=x^2y^2$. Find a formula for $y$ in the solution curve passing through $(1,1)$.

Question

A function $y$ satisfies the differential equation $y'=x^2y^2$. Find a formula for $y$ in the solution curve passing through $(1,1)$.

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How would I set up an equation? How should I start? Thank you.

Answer 1

This DE is separable. Another way of writing "passing through point (1,1)" is $y(1)=1$. As the DE is separable, we can integrate $$\int\frac{dy}{y^2}=\int x^2dx$$ to obtain $$-\frac{1}{y}=\frac{1}{3}x^3+c$$ where $c$ is the constant of integration. Now apply the condition $y(1)=1$ to obtain the constant. $$-1=\frac{1}{3}+c\implies c=-\frac{4}{3}$$ So the solution is $$y=\frac{3}{4-x^3}$$