5.9 KiB
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 nonhomogenous.
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 coeffictions, we assume solutions of the following form:
y(t) = e^{rt}
Substituting this equation into the differential equationm, we see the following:
e^{rt}(ar^2 + br + c) = 0
Thus, we allow the charactaristic 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 charactaristic 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 charactaristic 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 colliloquy 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 charactaristic 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 - Nonhomogenous 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 equivilent homogenous differential equation is
y'' + p(t) y' + q(t) y = 0
Theorem. Assume Y_1(t), Y_2(t) are solutions to the nonhomogenous 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 equivilent 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