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We have $$\frac{dy}{dx}=8y^2x^3, \ y(2)=0. $$ We run into problems when we try to find the value of the constant after integrating due to the initial values we are given. Does this mean that the only solution to the ODE is the constant solution $y=0$?

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Yes.

In fact, one may argue that since $y \equiv 0$ is a solution, and that the function $(x,y) \mapsto 8 y^2 x^3$ is real analytic (and hence at least Lipschitz continuous in both $x$ and $y$), by the Picard-Lindelof Theorem the differential equation admits unique solutions, that $y \equiv 0$ is the only solution.

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