Solve the differential equation $\dfrac{d^{2}y}{dx^{2}}=12x^{2}$ with the boundary conditions when $x=0,$ then $y=1$ and when $x=3,$ then $y=8$.

Solution All the antiderivatives of $\dfrac{d^{2}y}{dx^{2}}=12x^{2}$ are $$\hspace{12pt}\dfrac{dy}{dx}=4x^{3}+C_{1}$$

All the antiderivatives of $\dfrac{dy}{dx}=4x^{3}+C_{1}$ are $$y=x^{4}+C_{1}x+C_{2}$$

This is the general solution of the differential equation. To find $C_{1}$ and $C_{2}$ and the particular solution to the differential equation, we use the boundary conditions.

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The particular solution with the given boundary conditions is $$y=x^{4}-\dfrac{74}{3}x+1$$