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

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

Variation of Parameters for Higher Order Linear DEs

1. Identify the homogeneous solution to the $n^{\text{th}}$-order ODE, i.e., $y_h(t)=c_1y_1(t)+c_2y_2(t)+\cdots+c_ny_n(t)$ for linearly independent set of solutions $\left\{y_1(t),y_2(t),\ldots,y_n(t)\right\}$.
2. The particular solution is $y_p(t)=v_1(t)y_1(t)+v_2(t)y_2(t)+\cdots+v_ny_n(t)$ for some functions $v_1(t)$, $v_2(t)$, $\ldots$, and $v_n(t)$.
3. Find the Wronskian $W[y_1,\ldots y_n](t)$.
4. Find the Wronskian $W_k=(-1)^{n-k}W[y_1,\ldots,y_{k-1},y_{k+1},\ldots, y_n](t)$, for $k=1,\ldots,n$ NOTE: The $y_k$ column and the last row are deleted before calculating the Wronskian on the right side of the definition of $W_k$].
5. Then the particular solution is $\displaystyle y_p(t)=\sum_{k=1}^{n}\left(y_k(t)\int{\frac{g(t)W_k(t)}{aW[y_1,\ldots y_n](t)}\,dt}\right)$

Investigation 03

Use the method of variation of parameters to find the general solution to the following DEs:

1. $\displaystyle y^{\,\prime\prime\prime}-2y^{\,\prime\prime}+y^{\,\prime}=1$
2. $\displaystyle y^{\,\prime\prime\prime}+y^{\,\prime}=\tan{(x)}$
3. $\displaystyle y^{\,\prime\prime\prime}-2y^{\,\prime\prime}-y^{\,\prime}+2y=e^{3x}$

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Last Modified: Thursday, 8 October 2020 7:38 EDT