Method of factoring and reduction of order

Section 2.2 Method of factoring and reduction of order

In this section, we consider second-order linear homogeneous ODEs in the form

\begin{equation} a_2(t)y''+a_1(t)y'+a_0(t)y=0.\tag{2.2.1} \end{equation}

To form a FSS to (2.2.1) on an interval \(I\text{,}\) we need two linearly independent solutions \(y_1\) and \(y_2\) such that \(y(t)=c_1 y_1(t)+c_2 y_2(t)\) is a general solution. We introduce two substitution methods to find a FSS to (2.2.1): a substitution resulting from factoring a differential operator ("method of factoring") and a technique that can be used to ``vary a parameter" to obtain a solution \(y_2\) independent from a known solution \(y_1\) ("reduction of order").

Algorithm 2.2.1. Method of factoring.

Using differential operator notation, recall that (2.2.1) can also be written as

\begin{equation*} a_2(t)D^2y+a_1(t)Dy+a_0(t)y=0. \end{equation*}

Now we would like a factorization of the form

\begin{equation*} \left(b_1(t)D+b_2(t)\right)\left(b_3(t)+b_4(t)D\right)y=0. \end{equation*}

By substituting \(u=\left(b_3(t)D+b_4(t)\right)\text{,}\) we can solve a first-order ODE to solve for \(u\text{.}\) Then as a result, we can use this substitution to solve a first-order ODE for \(y\text{.}\)

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Example 2.2.2.

Factoring the differential operator \(ty''+(1-2t)y'+(t-1)y=0\) on \((0,\infty)\) can be written as \(tD^2y+(1-2t)Dy+(t-1)y=(D-1)(tD-t)y=0\) \((0,\infty)\text{.}\)

Let \(u=(tD-t)y\text{.}\) Use this substitution to solve a differential equation in terms of \(u\text{.}\)

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Use your result above to solve for \(y\text{.}\)

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Example 2.2.3.

Let \(g(t)=t^2\text{.}\)

Apply the differential operator \(D(tD-2)\) to \(g\text{.}\)

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Apply the differential operator \((tD-2)D\) to \(g\text{.}\)

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What can you conclude about differential operators in general?

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Now we assume we know one solution \(y_1\) to (2.2.1) on some interval \(I\text{.}\) We need a second solution \(y_2\) that is linearly dependent of \(y_1\) so that we obtain a FSS for (2.2.1). Recall that if \(z(t)=cy_1(t)\text{,}\) then \(z\) is a solution to (2.2.1) by the superposition principle Theorem 2.1.9.

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Now suppose \(v(t)=u(t)y_1(t)\text{.}\) We picked this function \(v\) by replacing the constant \(c\) in in \(z\) with an unknown function \(v\text{.}\) This is called varying the parameter. By construction, \(u\) is linearly independent of \(y_1\) on \(I\text{.}\) However independence alone is not sufficient to have a FSS. So we need to pick the function \(v\) that makes \(u\) into a solution of (2.2.1) on \(I\text{.}\)

Algorithm 2.2.4. Reduction of order.

Let \(y_1\) be a known solution to (2.2.1) on \(I\text{.}\)

Suppose \(y_2(t)=u(t)y_1(t)\) is another solution to (2.2.1) on \(I\text{.}\)

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Substitute \(y_2\) into (2.2.1).

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Regroup terms by \(u\text{.}\) Note that the term multiplied to \(u\) must vanish.

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Make a change of variables that allows rewriting the differential equation into a first-order linear separable ODE in terms of the variable \(w=u'\text{.}\)

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Solve the ODE for \(w\text{.}\)

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Integrate with respect to \(t\) to get \(u\text{.}\) Note the constant of integration can be absorbed.

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Substitute \(u\) into our starting point for \(y_2\) in Step 1. to complete the solution.

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Example 2.2.5.

Use reduction of order to find a second linearly independent solution to the differential equation \(y''-2y'+y=0\text{,}\) where \(y_1(t)=e^t\) is a known solution on \((-\infty,\infty)\text{.}\) Write a general solution.

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Example 2.2.6.

Use reduction of order to find a second linearly independent solution to the differential equation \(t^2y''-3ty'+4y=0\) on \((0,\infty)\) where \(y_1(t)=t^2\) is a known solution. Write a general solution.

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Example 2.2.7.

Use reduction of order to find a second linearly independent solution to the differential equation \(ty''+(1-2t)y'+(t-1)y=0\) on \((0,\infty)\) where \(y_1(t)=e^t\) is a known solution. Write a general solution.

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