# Variation of Parameters

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

• Section 4.6 -- Variation of Parameters

## Expected Educational Results

• Objective 13–1: I understand when variation of parameters is required to find particular solutions to nonhomogeneous equations.
• Objective 13–1: I can use variation of parameters to find particular solutions to nonhomogeneous equations.

## Variation of Parameters

### Definition: Particular Solution

A solution to a nonhomogeneous DE is called the particular solution, $\displaystyle y_p(t)$.

### Definition: Variation of Parameters

Variation of parameters is a more general method to find $y_p(t)$.

Variation of parameters is used for nonhomogeneous solutions when $f(t)$ contains factors or terms other than $P_m(t)$, $e^{rt}$, $\cos{(\beta t)}$, and $\sin{(\beta t)}$.

### Definition: General Solution

$\displaystyle y_g(t)=y_h(t)+y_p(t)$.

### Method of Variation of Parameters

NOTE: The terms determined by the method of variation of parameters must be linearly independent of the terms in $y_h(t)$.

1. Find two linearly independent solutions $\left\{y_1(t),y_2(t)\right\}$ to the corresponding homogeneous equation.

2. Let $y_p(t)=v_1(t)y_1(t)+v_2(t)y_2(t)$, for some functions $v_1(t)$ and $v_2(t)$.

3. Solve for $v_1(t)$:

• $v_1(t)=\int{\dfrac{-f(t)y_2(t)}{a W[y_1,y_2](t)}\,dt}$
4. Solve for $v_2(t)$:

• $v_2(t)=\int{\dfrac{f(t)y_1(t)}{a W[y_1,y_2](t)}\,dt}$
5. The particular solution is $y_p(t)=v_1(t)y_1(t)+v_2(t)y_2(t)$.

6. Write the general solution, $y(t)=y_h(t)+y_p(t)$.

#### Investigation 02

Solve the following DEs:

1. $\displaystyle y^{\,\prime\prime}+4y^{\prime}+4y=\sin{(x)}$
2. $\displaystyle y^{\,\prime\prime}+y^{\prime}=\sec{(x)}$
3. $\displaystyle y^{\,\prime\prime}+4y^{\prime}+4y=x^{-2}e^{-2x}$, $x>0$
4. $\displaystyle y^{\,\prime\prime}+4y=\tan{(2x)}$
5. $\displaystyle y^{\,\prime\prime}-2y^{\prime}+2y=e^x\sec{(x)}$
6. $\displaystyle y^{\,\prime\prime}+2y^{\prime}+y=e^{-x}\ln{(x)}$
7. $\displaystyle y^{\,\prime\prime}(t)-2y^{\,\prime}(t)+y(t)=\frac{e^t}{1+t^2}$