Solving a Second-Order Differential Equation

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.

When \(x=0,\qquad 1=0^{4}+C_{1}( 0) +C_{2}\qquad \hbox{so}\quad C_{2}=1\)

When \(x=3,\qquad 8=3^{4}+3C_{1}+1\qquad \hbox{so}\quad C_{1}=-\dfrac{74}{3}\)

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