Summary of Techniques: Solving Second Order Differential Equations

Summary of Techniques for Solving Second Order Differential Equations

We will now summarize the techniques we have discussed for solving second order differential equations.

• Real and Distinct Roots of The Characteristic Equation: If we have a second order linear homogneous differential with constant coefficients $a \frac{d^2 y}{dt^2} + b \frac{dy}{dt} + cy = 0$, and if the roots of the characteristic equation $ar^2 + br + c = 0$ are real and distinct, then the general solution for this differential equation is given by:

(1)

\begin{align} \quad y = Ce^{r_1t} + De^{r_2t} \end{align}

• Complex Roots of The Characteristic Equation: If we have a second order linear homogeneous differential with constant coefficients $a \frac{d^2 y}{dt^2} + b \frac{dy}{dt} + cy = 0$, and if the roots of the characteristic equation $ar^2 + br + c = 0$ are complex (conjugates of each other), then $r_1 = \lambda + \mu i$ and $r_2 = \lambda - \mu i$ for some $\lambda, \mu \in \mathbb{R}$ and the general solution for this differential equation is given by:

(2)

\begin{align} \quad y = Ce^{\lambda t} \cos (\mu t) + De^{\lambda t} \sin (\mu t) \end{align}

• Repeated Roots of The Characteristic Equation: If we have a second order linear homogeneous differential with constant coefficients $a \frac{d^2 y}{dt^2} + b \frac{dy}{dt} + cy = 0$, and if the roots of the characteristic equation $ar^2 + br + c = 0$ are real and not distinct (that is $r_1 = r_2$) then the general solution for this differential equation is given by:

(3)

\begin{align} \quad y = Ce^{r_1t} + Dte^{r_1t} \end{align}

• Reduction of Order on Second Order Linear Homogenous Differential Equations: If we have a second order linear homogeneous differential equation $\frac{d^2 y}{dt^2} + p(t) \frac{dy}{dt} + q(t)y = 0$ and we know that $y = y_1(t)$ is a nonzero solution to our differential equation, then we can assume that $y = v(t) v_1(t)$ is a solution to our differential equation and plug it into our differential equation to obtain a first order differential equation for the function $v'(t)$ for which we can apply techniques of solving first order differentials to obtain $v'(t)$, integrate to get $v(t)$, and then obtain a second solution $y_2(t) = v(t) y_1(t)$.

(4)

\begin{align} \quad (2y_1'(t) + p(t)y_1(t))v'(t) + y_1(t)v''(t) = 0 \end{align}

• Euler Differential Equations: A second order linear homogeneous differential equation in the form $t^2 \frac{d^2 y}{dt^2} + \alpha t \frac{dy}{dt} + \beta y = 0$ for some $\alpha, \beta \in \mathbb{R}$ is called an Euler differential equation. If we let $x = \ln t$, then we can transform this differential equation into the following differential equation with constant coefficients and solve for $y$ in terms of $x$ by using one of the techniques for solving linear homogeneous differential equations with constant coefficients. We can then obtain a solution $y$ in terms of $t$ by substituting back $x = \ln t$.

(5)

\begin{align} \quad \frac{d^2 y}{dx^2} + (\alpha - 1) \frac{dy}{dx} + \beta y = 0 \end{align}

• The Method of Undetermined Coefficients: If we have a second order linear nonhomogeneous differential equation with constant coefficients, then if the function $g(t)$ is particularly nice - that is, $g(t)$ contains only exponential functions, sines, cosines, or polynomials, then we can assume a form for a particular solution $Y(t)$ - plug this into our differential equation, and then solve for the coefficients. The general solution to this differential equation is then $y(t) = y_h(t) + Y(t)$ where $y_h(t)$ is the solution to the corresponding second order linear homogeneous differential equation.

• The Method of Variation of Parameters: If we have a second order linear nonhomogeneous differential equation with constant coefficients and if $g(t)$ is not suitable for applying the method of undetermined coefficients, then instead, we can assume that the particular solution $Y(t) = u_1(t)y_1(t) + u_2(t)y_2(t)$ where $y_1(t)$ and $y_2(t)$ form a fundamental set of solutions to the corresponding second order linear homogeneous differential equation and where $u_1$ and $u_2$ can be determined by solving the system of equations below for $u_1'(t)$ and $u_2'(t)$ and integrating the results to obtain a particular solution. Once again, $y(t) = y_h(t) + Y(t)$.

(6)

\begin{align} \quad \left\{\begin{matrix} u_1'(t)y_1(t) + u_2'(t)y_2(t) = 0\\ u_1'(t)y_1'(t) + u_2'(t)y_2'(t) = g(t) \end{matrix}\right. \end{align}

We will also comment on the existence of solutions for second order linear differential equations and general solution sets to second order differential equations.

Theorem (Existence/Uniqueness of Solutions to Second Order Linear Differential Equations): Let $p$, $q$, and $g$ be continuous functions on an open interval $I$ such that $t_0 \in I$. Then the second order linear differential equation $\frac{d^2 y}{dt^2} + p(t) \frac{dy}{dt} + q(t) y = g(t)$ with the initial conditions $y(t_0) = y_0$ and $y'(t_0) = y'_0$ has a unique solution $y = \phi (t)$ throughout $I$.

Theorem (The Principle of Superposition): If $\frac{d^2 y}{dt^2} + p(t) \frac{dy}{dt} + q(t) y = 0$ is a second order linear differentiation equation and $y = y_1(t)$ and $y = y_2(t)$ are both solutions to this differential equation, then for $C$ and $D$ as constants, $y = Cy_1(t) + Dy_2(t)$ is also a solution.

Theorem (Abel's Identity): Let $\frac{d^2 y}{dt^2} + p(t) \frac{dy}{dt} + q(t) y = 0$ be a second order linear homogeneous differential equation where $p$ and $q$ are continuous on an open interval $I$ such that $t_0 \in I$. Then the Wronskian of $y_1$ and $y_2$ at some $t$ is given by $W(y_1, y_2) = C e^{- \int p(t) \: dt}$ where $C$ is some constant dependent on $y_1$ and $y_2$.

Theorem (Wronskian Determinants): Let $\frac{d^2 y}{dt^2} + p(t) \frac{dy}{dt} + q(t) y = 0$ be a second order linear homogeneous differential equation where $p$ and $q$ are continuous functions on an open interval $I$ such that $t_0 \in I$ and with the initial conditions $y(t_0) = y_0$ and $y'(t_0) = y'_0$. If $y = y_1(t)$ and $y = y_2(t)$ are solutions to this differential equation then there exists constants $C$ and $D$ for which $y = Cy_1(t) + Dy_2(t)$ is a solution to the initial value problem if and only if the Wronskian at $t_0$ is nonzero, that is $W(y_1, y_2) \biggr \rvert_{t_0} = y_1(t_0)y_2'(t_0) - y_1'(t_0)y_2(t_0) \neq 0$.

Theorem (Fundamental Solutions): Let $\frac{d^2 y}{dt^2} + p(t) \frac{dy}{dt} + q(t)y = 0$ be a second order linear homogeneous differential equation where $p$ and $q$ are continuous on an open interval $I$ such that $t_0 \in I$, and let $y = y_1(t)$ and $y = y_2(t)$ be two solutions to this differential equation. The set of all linear combinations of these two solutions, $y = Cy_1(t) + Dy_2(t)$ where $C$ and $D$ are constants contains all solutions to this differential equation if and only if there exists a point $t_0$ for which the Wronksian of $y_1$ and $y_2$ at $t_0$ is nonzero, that is $W(y_1, y_2) \biggr \rvert_{t_0} \neq 0$.