3.8 Non-homogenous Differential Equations

Second-Order, Linear, Non-homogenous Differential Equation Form

The non-homogenous aspect comes from the fact that there is now a forcing function in the form of g(t); with homogenous equations, however, we only ever have 0 on the right-hand side.

Fundamental Theorem for Non-Homogenous Equations

If Y₁(t) and Y₂(t) are solutions to the non-homogenous form of a differential equation and y₁(t) and y₂(t) are the solutions to the homogenous form of the differential equation, then we have:

Y₁(t) - Y₂(t) = c₁y₁(t) + c₁y₂(t)

General Solution to a Non-Homogenous Equation

The solutions to Non-homogenous Equations will always be written in the form of:

y(t) = y_c(t) = Y_P(t)

y_c(t) — the complementary solution that acts as the solution to the homogenous form of the differential equation

Y_P(t) — the particular solution that acts as the solution to the non-homogenous differential equation.

The final (general) solution is the linear combination of the two solutions themselves.

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