Second Order Differential Equations

Solving differential equations with second derivatives using a coupled system technique.

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Second order differential equations

AHL AI 5.18

A second order differential equation is a differential equation of the form

dtd2x=f(x,dtdx,t)

To solve these equations, we rewrite it as a system of coupled first order equations. Using the fact that dtd2x=dtd(dtdx) and substituting dtdx=y,

⎩⎪⎨⎪⎧dtdx=ydtdy=f(x,y,t)

This equation can be solved with the usual techniques for coupled systems. On exams, questions involving second-order differential equations are usually set in a real-world context such as movement.