# The Method of Undetermined Coefficients

Author: John J Weber III, PhD Corresponding Textbook Sections:

• Section 4.4 – Nonhomogeneous Equations: The Method of Undetermined Coefficients
• Section 4.5 – The Superposition Principle and Undetermined Coefficients Revisited

## Expected Educational Results

• Objective 12–1: I can find particular solutions to nonhomogeneous equations.
• Objective 12–2: I can find general solutions to nonhomogeneous equations.
• Objective 12–2: I can solve higher-order nonhomogeneous IVPs.

## Method of Undetermined Coefficients

### Initial Value Problems [IVPs]

To solve IVPs:

1. Find $\displaystyle y_h(t)$.
2. Find $\displaystyle y_p(t)$.
3. Find coefficients for $\displaystyle y_p(t)$.
4. Use $\displaystyle y_g(t)=y_h(t)+y_p(t)$ to find coefficients in $\displaystyle y_h(t)$.

NOTE: To find coefficients $c_i$ in $y_h(t)$ for an IVP, you must use $y_g(t)$.

#### Investigation 04

Solve the following IVPs:

1. $\displaystyle y^{\,\prime\prime}-6y^{\,\prime}-7y=-9e^{-2x}$, $y(0)=-2$, $y^{\,\prime}(0)=-13$
2. $\displaystyle y^{\,\prime\prime}-2y^{\,\prime}+4y=8x-12\sin{(2x)}$, $y(0)=-2$, $y^{\,\prime}(0)=8$