4.2 Second-Order Linear ODEs

Second-order linear ODEs are key in mathematical physics. They model everything from simple harmonic motion to complex wave phenomena. Understanding their classification and solutions is crucial for tackling real-world problems.

This section covers how to classify these ODEs, find general solutions for homogeneous cases, and handle non-homogeneous equations. We'll also look at initial value problems and apply these concepts to physical systems like oscillators.

Classification and General Solutions of Second-Order Linear ODEs

Classification of second-order linear ODEs

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• General form of a second-order linear ODE with constant coefficients is $ay'' + by' + cy = f(x)$ where $a$, $b$, and $c$ are constants with $a \neq 0$ and $f(x)$ is a function of $x$
• Homogeneous ODEs have $f(x) = 0$ while non-homogeneous ODEs have $f(x) \neq 0$
• Characteristic equation $ar^2 + br + c = 0$ determines the type of solution based on its roots
  • Real and distinct roots lead to a solution of the form $y = c_1e^{r_1x} + c_2e^{r_2x}$ ($r_1 \neq r_2$)
  • Real and repeated roots result in a solution of the form $y = (c_1 + c_2x)e^{rx}$
  • Complex conjugate roots $\alpha \pm i\beta$ give a solution of the form $y = e^{\alpha x}(c_1\cos(\beta x) + c_2\sin(\beta x))$

General solutions for homogeneous ODEs

• Find the roots of the characteristic equation $ar^2 + br + c = 0$
• Construct the general solution based on the type of roots obtained
  • For real and distinct roots $r_1$ and $r_2$, the solution is $y = c_1e^{r_1x} + c_2e^{r_2x}$
  • For real and repeated roots $r$, the solution is $y = (c_1 + c_2x)e^{rx}$
  • For complex conjugate roots $\alpha \pm i\beta$, the solution is $y = e^{\alpha x}(c_1\cos(\beta x) + c_2\sin(\beta x))$ where $\alpha$ and $\beta$ are the real and imaginary parts of the roots

Particular solutions for non-homogeneous ODEs

• Use the method of undetermined coefficients to find particular solutions of non-homogeneous ODEs
• Identify the form of the particular solution based on the non-homogeneous term $f(x)$
  1. For polynomial $f(x)$ of degree $n$, use $y_p = A_nx^n + A_{n-1}x^{n-1} + \cdots + A_1x + A_0$
  2. For exponential $f(x) = e^{\alpha x}$, use $y_p = Ae^{\alpha x}$
  3. For trigonometric $f(x) = \cos(\beta x)$ or $\sin(\beta x)$, use $y_p = A\cos(\beta x) + B\sin(\beta x)$
• Substitute the particular solution into the ODE and solve for the unknown coefficients ($A_n, A_{n-1}, \ldots, A_1, A_0, A, B$)
• The general solution is the sum of the homogeneous solution $y_h$ and the particular solution $y_p$, i.e., $y = y_h + y_p$

Initial value problems in ODEs

• Determine the general solution by finding both the homogeneous and particular solutions
• Apply the given initial conditions to find the values of arbitrary constants $c_1$ and $c_2$
  1. Substitute the initial conditions into the general solution and its first derivative
  2. Solve the resulting system of equations for the arbitrary constants

Applications of ODEs in physics

• Model simple harmonic oscillators using the ODE $my'' + ky = 0$ where $m$ is the mass and $k$ is the spring constant
  • The solution is $y = A\cos(\omega t) + B\sin(\omega t)$ with angular frequency $\omega = \sqrt{\frac{k}{m}}$
• Describe damped harmonic oscillators using the ODE $my'' + cy' + ky = 0$ where $c$ is the damping coefficient
  • The solution depends on the roots of the characteristic equation
    1. Overdamped (real, distinct, negative roots): $y = c_1e^{r_1t} + c_2e^{r_2t}$
    2. Critically damped (real, repeated, negative roots): $y = (c_1 + c_2t)e^{-\frac{c}{2m}t}$
    3. Underdamped (complex conjugate roots): $y = e^{-\frac{c}{2m}t}(c_1\cos(\omega t) + c_2\sin(\omega t))$ with $\omega = \sqrt{\frac{4mk-c^2}{4m^2}}$
• Analyze forced harmonic oscillators described by the ODE $my'' + cy' + ky = F(t)$ where $F(t)$ is the external forcing function
  • The solution is the sum of the homogeneous solution and a particular solution depending on the form of $F(t)$