How to Use the Reduction of Order Method for Differential Equations

If you have a differential equation without $y$ , but also only derivatives of $y$ , then you can use substitution to reduce the order of the equation. You put $u = y^{'}$ , which gives you $u^{'} = y^{″}$ and $u^{″} = y^{‴}$ . This gives you a lower-order differential equation, which you can then solve using the usual methods.

Example 1

Solve the differential equation $y^{‴} - 7 y^{″} + 12 = 0$

Let $y^{'} = u$ and substitute:

u^{″} - 7 u^{'} + 12 = 0 .

This has the characteristic equation

\begin{array}{l} r^{2} - 7 r + 12 & = 0 \\ (r - 4) (r - 3) & = 0 \end{array}

which has the solutions $r_{1} = 4$ and $r_{2} = 3$ . Enter $r_{1}$ and $r_{2}$ into the formula for the solution of the characteristic equation and get

u (x) = C_{1} e^{4 x} + C_{2} e^{3 x}

You now need to substitute back $y^{'} = u$ :

y^{'} = C_{1} e^{4 x} + C_{2} e^{3 x}

By integrating, you get

\begin{array}{l} y & = \int y^{'} d x \\ = \int C_{1} e^{4 x} + C_{2} e^{3 x} d x \\ = \frac{C_{1}}{4} e^{4 x} + \frac{C_{2}}{3} e^{3 x} + C_{3} \end{array}

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The Differential Equation for Harmonic Oscillators

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