Solving Second Order Ordinary Differential Equations

Homogenous

$$\frac{\text{d}^2y}{\text{d}x^2}+ a\frac{\text{d}y}{\text{d}x}+ +by =0\,;\;a,b\,\text{are constants}$$

Consider the function $y=e^{mx}$. Here $m$ is a constant to be found.

By applying the function to the above equation:

$$m^2 + am + b = 0$$

The above equation is called the Auxiliary equation or Characteristic equation.

Case 1: Distinct real roots

$$y = Ae^{m_1x}+Be^{m_2x}$$

Case 2: Equal real roots

$$y = (Ax+B)e^{mx}$$

Case 3: Complex conjugate roots

$$y = Ae^{(p+iq)x} + Be^{(p−iq)x} = e^{px}\big(C\cos{(qx)}+D\sin{(qx)}\big)$$

Non-homogenous

$$\frac{\text{d}^2y}{\text{d}x^2}+ a\frac{\text{d}y}{\text{d}x}+ +by =q(x)\,;\;a,b\,\text{are constants}$$

Method of undetermined coefficients

$y_p$ is guessed with undetermined coefficients. And subtituted in the given equation to find the coefficients. The guess depends on the nature of $q(x)$.

If $q(x)$ is:$$

• a constant, $y_p$ is a constant
• $kx$, $y_p=ax+b$
• $kx^2$, $y_p=ax^2+bx+c$
• $k\sin{x}$ or $k\cos{x}$, $y_p=a\sin{x}+b\cos{x}$
• $e^{kx}$, $y_p=ce^{kx}$ (Only works if $k$ is not a root of auxiliary equation)
• A product of $e^{kx}$ and some $f(x)$, guess $y_p$ for $f(x)$ individually, and then multiply by $e^{kx}$ (without coefficients)
• A product of polynomials and trig functions, guess $y_p$ for the polynomial, and multiply that by the appropriate cosine. Then add on a new guess for the polynomial with different coefficients and multiply that by the appropriate sine.
• A sum of functions, can be guessed individually and be summed up

Steps

• Solve for $y_c$
• Based on the form of $q(x)$ , make an initial guess for $y_p$.
• Check if any term in the guess for $y_p$ is a solution to the complementary equation.
• If so, multiply the guess by $x$. Repeat this step until there are no terms in $y_p$ that solve the complementary equation.
• Substitute $y_p$ into the differential equation and equate like terms to find values for the unknown coefficients in $y_p$.
• If coefficients were unable to be found (they cancelled out or something like that), multiply the guess by $x$ and start again.
• $y=y_p+y_c$