Section 3 - Second Order Differential Equations

Section 3.1 - Basic Concepts

This section is from Paul's Online Math Notes.

All second-order differential equations can be written in the following form:

\[ p(t) y'' + q(t) y' + r(t) y = g(t) \]

In the case where \(p(t)\), \(q(t)\), and \(r(t)\) are constants, we write the equation as the following:

\[ ay'' + by' + cy = g(t) \]

This is a second-order differential equation with constant coefficients.

Definition. In the event that \(g(t) = 0\), we say the equation is homogenous. Otherwise, the equation is nonhomogeneous.

Definition. Principal of Superposition. Let \(y_1(t)\) and \(y_2(t)\) be solutions to a linear, homogenous differential equation. Then, any linear combination of said solutions is also a solution to the differential equation. In other words, with \(c_1, c_2 \in \mathbb{R}\), the following is a solution to a differential equation.

\[ y(t) = c_1 y_1(t) + c_2 y_2(t) \]

Given a second-order homogenous differential equation with constant coefficients, we assume solutions of the following form:

\[ y(t) = e^{rt} \]

Substituting this equation into the differential equation, we see the following:

\[ e^{rt}(ar^2 + br + c) = 0 \]

Thus, we allow the characteristic equation of the differential equation to be as follows:

\[ ar^2 + br + c = 0 \]

Section 3.2 - Real & Distinct Roots

This section is from Paul's Online Math Notes.

When the two roots to the characteristic equation are discrete roots in the real numbers, we see the following solutions.

\[ y_1(t) = e^{r_1 t} \]

\[ y_2(t) = e^{r_2 t} \]

Thus,

\[ y(t) = c_1 e^{r_1 t} + c_2 e^{r_2 t} \]

Section 3.3 - Complex Roots

This section is from Paul's Online Math Notes.

Let the solutions to the characteristic equation be of the following form:

\[ r_{1,2} = \lambda \pm \mu i \]

Thus, our two solutions are

\[ y_1(t) = e^{(\lambda + \mu i)t} \]

\[ y_2(t) = e^{(\lambda - \mu i)t} \]

Recall Euler's Formula:

\[ e^{i \theta} = \cos \theta + i \sin \theta \]

A corollary of Euler's formula is the following:

\[ e^{-i \theta} = \cos(-\theta) + i \sin(-\theta) = \cos \theta - i \sin \theta \]

Thus, we can write our solutions as the following:

\[\begin{align} y_1(t) &= e^{(\lambda + \mu i)t} &= e^{\lambda t} e^{i \mu t} &= e^{\lambda t}(\cos(\mu t) + i \sin(\mu t)) \\ y_2(t) &= e^{(\lambda - \mu i)t} &= e^{\lambda t} e^{-i \mu t} &= e^{\lambda t}(\cos(\mu t) - i \sin(\mu t)) \end{align}\]

A linear combination of the two solutions can be written as the following:

\[ y(t) = c_1 e^{\lambda t} \cos(\mu t) + c_2 e^{\lambda t} \sin(\mu t) \]

Section 3.4 - Repeated Roots

This section is from Paul's Online Math Notes.

Assume the solutions to the characteristic equations are \(r = r_1 = r_2\). Thus, the two equations \(y_t(t)\) and \(y_2(t)\) are not linearly independent.

After a lot of algebra, we see that

\[y_1(t) = e^{rt}\]

\[y_2(t) = t e^{rt}\]

Section 3.5 - Reduction of Order

This section is from Paul's Online Math Notes.

Skipped.

Section 3.6 - Fundamental Set of Solutions, Wronskian

This section is from Paul's Online Math Notes.

Definition. Given two functions \(f(t)\), \(g(t)\), the Wronskian is defined as

\[ W(f,g) = \det \begin{vmatrix} f(t) & g(t) \\ f'(t) & g'(t) \end{vmatrix} \]

Definition. If \(W(f, g) \neq 0\), then \(f(t)\) and \(g(t)\) are said to form a fundamental set of solutions, and can be superimposed to form the general solution.

Section 3.8 - Nonhomogeneous Differential Equations

This section is from Paul's Online Math Notes.

Assume we have the differential equation as follows:

\[ y'' + p(t) y' + q(t) y = g(t) \]

The equivalent homogenous differential equation is

\[ y'' + p(t) y' + q(t) y = 0 \]

Theorem. Assume \(Y_1(t)\), \(Y_2(t)\) are solutions to the nonhomogeneous differential equations. Then, \(Y_1(t) - Y_2(t)\) is a solution to the homogenous differential equation. This can be proved by substitution.

Thus, with \(y_h(t)\) the solution to the homogenous problem, and \(y_p(t)\) the solution to this particular problem, we can say that the general form of the solution to this differential equation is

\[ y(t) = y_h(t) + y_p(t) \]

Section 3.9 - Undetermined Coefficients

This section is from Paul's Online Math Notes.

We know the following guesses for functions.

\(g(t)\)	\(y_p\) guess
\(\alpha e^{\beta t}\)	\(A e^{\beta t}\)
\(a \cos(\beta t)\)	\(A \cos(\beta t) + B \sin(\beta t)\)
\(b \sin(\beta t)\)	\(A \cos(\beta t) + B \sin(\beta t)\)
\(a \cos(\beta t) + \sin(\beta t)\)	\(A \cos(\beta t) + B \sin(\beta t)\)
n-th degree polynomial	\(A_nt^n + A_{n-1}t^{n-1} + A_1 t + A_0\)

Combine this with the following:

Theorem. Given \(y_{p_1}(t)\) is a solution to \(y'' + p(t)y' + q(t)y = g_1(t)\) and \(y_{p_2}(t)\) is a solution to \(y'' + p(t)y' + q(t)y = g_2(t)\), then the function \(y_{p_1}(t) + y_{p_2}(t)\) is a solution to \(y'' + p(t)y' + q(t)y = g_1(t) + g_2(t)\)

Section 3.10 - Variation of Parameters

This section is from Paul's Online Math Notes.

Assume we have the differential equation as follows:

\[ y'' + p(t) y' + q(t) y = g(t) \]

The equivalent homogenous differential equation is

\[ y'' + p(t) y' + q(t) y = 0 \]

For this method, we must have \(y_1(t)\) and \(y_2(t)\) known. Through a lot of math, we see that

\[ y_p = -y_1 \int \frac{y_2(t)g(t)}{W(y_1, y_2)} dt + y_2 \int \frac{y_1(t)g(t)}{W(y_1, y_2)} dt \]