3.5 Reduction of Order

Reduction of Order

When given a second-order differential equation whose coefficients are non-constant, solution functions can be found; however, applying this method requires that we already know one solution. If we do not know one solution, then we cannot apply this method whatsoever.

Reduction of Order Algorithm

Given a solution function f(t), we can solve for the other solution to the 2nd-order differential equation by realizing that its second solution would have to be in the form of y₂(t) = v(t)f(t) — where v(t) is some non-constant function; recall that v(t) cannot be a constant because then y₂(t) and y₁(t) would not be linearly independent of each other.

From here, we calculate the first and second derivatives of y₂(t) and plug them into the original differential equation. From here, we will always obtain a second-order differential equation with just the y” and y’ term (no y term).

From here, we do a substitution so as to reduce the second-order into a first-order (hence the name of the algorithm).

Plug the substitutions back into the original equation and now we have a first-order linear differential equation.

Reduction of Order Example: Part 1

Reduction of Order Example: Part 2

Reduction of Order Example: Part 3

Reduction of Order Example: Part 4

Reduction of Order Formula (Alternative Approach)

This is a generalization of the algorithmic approach utilized for this topic so as to not have to re-invent the wheel every single time: