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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 7y + 12 = 0

Let y = u and substitute:

u 7u + 12 = 0.

This has the characteristic equation r2 7r + 12 = 0 (r 4)(r 3) = 0

which has the solutions r1 = 4 and r2 = 3. Enter r1 and r2 into the formula for the solution of the characteristic equation and get

u(x) = C1e4x + C 2e3x

You now need to substitute back y = u:

y = C 1e4x + C 2e3x

By integrating, you get y =ydx =C1e4x + C 2e3xdx = C1 4 e4x + C2 3 e3x + C 3

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