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, $y(t)=y_h(t)+y_p(t)$.

Derivation of the Method of Variation of Parameters

Investigation 01

1. Conditions

   1. Suppose $y_h(t)=c_1y_1(t)+c_2y_2(t)$ is the homogeneous solution to $ay^{\,\prime\prime}+by^{\,\prime}+cy=f(t)$, $a\ne 0$.
   2. Let $y_p(t)=u_1(t)y_1(t)+u_2(t)y_2(t)$ for some functions $u_1(t)$ and $u_2(t)$.
   3. Let $u_1^{\,\prime}(t)y_1(t)+u_2^{\,\prime}(t)y_2(t)=0$ [this condition is a simplifying assumption].

2. We need to show Equation (\ref{eq:v1}) and Equation (\ref{eq:v2}) above are consequences of the two conditions.

   1. Find $y_p^{\,\prime}(t)$. Use the simplifying assumption to simplify $y_p^{\,\prime}(t)$. Explain why the simplifying assumption is helpful.

   2. Find $y_p^{\,\prime\prime}(t)$.

   3. Substitute $y_p(t)$, $y_p^{\,\prime}(t)$, and $y_p^{\,\prime\prime}(t)$ into $ay^{\,\prime\prime}+by^{\,\prime}+cy=f(t)$.

   4. Expand the expression on the left side of the equation.

   5. Since we are interested in finding functions, $u_1(t)$ and $u_2(t)$, factor the expression using common factors, $u_1(t)$ and $u_2(t)$.

   6. What do you notice about the factors multiplying $u_1(t)$ and $u_2(t)$? Explain.

   7. Divide through by the coefficient $a$.

   8. We now have two equations with two unknowns, $u_1^{\,\prime}(t)$ and $u_2^{\,\prime}(t)$:

      1. The equation from Step vii.
      2. The simplifying assumption.

   9. Solve for $u_1^{\,\prime}(t)$ and $u_2^{\,\prime}(t)$.

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