Ordinary Differential Equations

In the previous section, we saw that the method of undetermined coefficients is a delicate method. In this section, we consider second-order linear nonhomogeneous ODEs of the form

where \(g\) is a continuous function of \(t\text{.}\) While the method of undetermined coefficients was an algebra ("bookkeeping") method, we now introduce a calculus method that applies to a wider range of linear nonohomogeneous ODEs.

Lemma 2.6.1. Cramer’s rule (2x2 case).

Consider the linear system

\begin{equation*} \left\{ \begin{array}{ll} ax+by=s \\ cx+dy=t. \end{array} \right. \end{equation*}

If \(ad-bc \neq 0\text{,}\) then the system has a unique solution given by

\begin{equation*} x = \dfrac{\mathrm{det}\begin{bmatrix} s&b\\ t&d\end{bmatrix}}{\mathrm{det}\begin{bmatrix}a&b \\ c&d\end{bmatrix}} \text{ and } y=\dfrac{\mathrm{det}\begin{bmatrix} a& s \\ c& t \end{bmatrix}}{\mathrm{det}\begin{bmatrix} a& b \\ c& d \end{bmatrix}} \end{equation*}

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Proof.

later!

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Recall from the superposition principle (Theorem Theorem 2.1.9) that any scalar multiple of a homogeneous solution to (2.6.1) on an interval \(I\) is also a solution on \(I\text{.}\) For (2.6.1), a homogeneous solution on \(I\) can be written as

To satisfy the right-hand side when \(g(t) \neq 0\text{,}\) we can "vary the parameters" to get

Theorem 2.6.2. Variation of parameters.

Suppose \(y_1\) and \(y_2\) represent the homogeneous solutions to

\begin{equation} y''+P(t)y'+Q(t)y=f(t)\tag{2.6.3} \end{equation}

on some interval \(I\text{.}\) Then the particular solution to (2.6.3) is of the form (2.6.2), where \(u_1(t)=-\displaystyle\int \dfrac{y_2(t)f(t)}{W\{y_1,y_2\}(t)} \mathrm{d}t\) and \(u_2(t)=\displaystyle\int \dfrac{y_1(t)f(t)}{W\{y_1,y_2\}(t)} \mathrm{d}t\text{.}\)

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Proof.

later!

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Algorithm 2.6.3. Process for variation of parameters.

1. Solve for \(y_h\text{.}\)
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2. Put the ODE into standard form. This step is necessary!
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3. Calculate the Wronskian. Once we have \(W\{y_1,y_2\}\text{,}\) we must stick with our choices of \(y_1\) and \(y_2\text{.}\)
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4. Assume \(y_p\) is in the form (2.6.2).
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5. Substitute \(y_p\) into the ODE in standard form.
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6. Use Cramer’s rule to solve \(u_i'\text{.}\)
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7. Integrate and substitute into \(y_p\text{,}\) combining any like terms.
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8. Check if any terms of \(y_p\) are absorbed by \(y_h\text{.}\)
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9. Write a general solution and evaluate initial conditions, if any.
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Example 2.6.4.

Consider the differential equation \(2y''-4y'+2y=\dfrac{4e^t}{t^2}\) with \(t>0\text{.}\)

1. Classify the differential equation by order, linearity, type of coefficients, and whether it is homogeneous or nonhomogeneous.
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2. What method(s) can you use to solve this differential equation?
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3. Find the general solution.
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Example 2.6.5.

Consider the differential equation \(t^2y''-ty'+y=\dfrac{3t}{\ln(t)}\) with \(t>1\text{.}\)

1. Classify the differential equation by order, linearity, type of coefficients, and whether it is homogeneous or nonhomogeneous.
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2. What method(s) can you use to solve this differential equation?
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3. Find the general solution.
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Example 2.6.6.

Consider the differential equation \(t^2y''-2ty'+2y=\dfrac{4t^3}{1+t^2}\) with \(t>0\text{.}\)

1. Classify the differential equation by order, linearity, type of coefficients, and whether it is homogeneous or nonhomogeneous.
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2. What method(s) can you use to solve this differential equation?
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3. Find the general solution.
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Example 2.6.7.

Let \(y''+y=f(t)\text{,}\) where \(f\) is a continuous function on \((-\infty,\infty)\text{.}\) Show that a general solution is of the form \(y(t)=c_1\cos(t)+c_2\sin(t)+\displaystyle\int_0^t f(s)\sin(t-s)\mathrm{d}s\text{.}\)

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