(Number 3 from Section 4.6 in the Differential Equations textbook by Nagle, Saff, and Snider)
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Handout Lesson 19, Variation of Parameters
Textbook Section(s).
This lesson is based on Section 3.5 of your textbook by Edwards, Penney, and Calvis.
Particular solutions for nonhomogeneous equations.
In the last lesson, we used the Method of Undetermined Coefficients to find particular solutions for nonhomogeneous linear differential equations with constant coefficients.
\begin{gather}
a_ny^{(n)}+a_{n-1}y^{(n-1)}+\dots+a_{1}y'+a_0y=f(x)\tag{✶}
\end{gather}
where \(a_0, a_1, \dots, a_{n-1}, a_n\) are all constants.
The Method of Undetermined Coefficients only worked when
Today, we continue to search for particular solutions of nonhomogeneous linear differential equations. We will learn a new method called Variation of Parameters. The method of Variation of Parameters:
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could, in theory, be used to find a particular solution for a differential equation of the form shown in (✶) when \(f(x)\) has the special form required by the Method of Undetermined Coefficients. In my experience, the Method of Undetermined Coefficients is usually easier to use in this case.
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can be used to find a particular solution for a differential equation of the form shown in (✶) when \(f(x)\) does NOT HAVE the special form required by the Method of Undetermined Coefficients.
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Can be used to find a particular solution for nonhomogeneous linear differential equation in which not all of the coefficients are constants. In other words, we are now looking for a particular solution of an equation of the form\begin{gather} y^{(n)}+p_1(x)y^{(n-1)}+\dots+p_{n-1}(x)y'+p_n(x)y=f(x)\tag{✶✶} \end{gather}
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Assumes that you have a set of \(n\) linearly independent solutions for the homogeneous differential equation associated with (✶✶). This is a BIG ASSUMPTION.
Variation of parameters for second-order, linear differential equations.
In this course, we will only study the method of Variation of Parameters for second-order, linear differential equations. The method of Variation of Parameters is used to find a particular solution of a linear differential equation in . Since we are studying second-order linear differential equations, we are concerned with equations of the form:
\begin{gather}
y''+P(x)y'+Q(x)y=F(x)\tag{#}
\end{gather}
Given: A general solution to the homogeneous equation associated with (#):
\begin{equation*}
y_c=c_1y_1+c_2y_2
\end{equation*}
where \(y_1\) and \(y_2\) are a pair of linearly independent functions.
Seek: A particular solution of (#) of the form
\begin{equation*}
y_p=u_1y_1+u_2y_2
\end{equation*}
where \(u_1\) and \(u_2\) are .
Because we are trying to find two unknowns ( ), we need 2 equations that contain the unknowns. One necessary equation is the equation in (#). There is no other necessary equation, so we can choose a second equation that will make life easier. Experience has shown mathematicians that the following equation will simplify our lives down the road:
\begin{gather}
u_1'y_1+u_2'y_2 = 0 \tag{##}
\end{gather}
Now use \(y_p=u_1y_1+u_2y_2\text{,}\) calculate \(y_p'\) and \(y_p''\text{,}\) and substitute into (#).
Summary of Variation of Parameters.
For the differential equation
\begin{gather}
y''+P(x)y'+Q(x)y=F(x)\tag{†}
\end{gather}
with general solution of the associated homogeneous equation
\begin{equation*}
y_c=c_1y_1+c_2y_2
\end{equation*}
where \(y_1\) and \(y_2\) are linearly independent functions, we can find a particular solution, \(y_p\text{,}\) of (†) have the form
\begin{equation*}
y_p(x)=u_1(x)y_1(x)+u_2(x)y_2(x) \quad \text{ or }
y_p=u_1y_1+u_2y_2
\end{equation*}
by
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solving the following system of equations for \(u_1'\) and \(u_2'\)\begin{equation*} \begin{cases} y_1u_1'+y_2u_2' = 0 \\ y_1'u_1'+y_2'u_2' = F(x) \\ \end{cases} \end{equation*}
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Construct your particular solution \(y_p\) of (†) as\begin{equation*} y_p=u_1y_1+u_2y_2 \end{equation*}
