Homogeneous equations with constant coefficients
Second-order linear equations model springs, circuits, and vibrations. With constant coefficients, they reduce to solving a quadratic.
The equation
A second-order linear equation with constant coefficients has the form \(ay'' + by' + cy = g(x)\). If \(g = 0\), it is homogeneous. The general solution of a homogeneous second-order equation is a combination \(c_1y_1 + c_2y_2\) of two independent solutions, with two constants fixed by \(y(x_0)\) and \(y'(x_0)\).
The characteristic equation
Try \(y = e^{rx}\). Substituting gives \(e^{rx}(ar^2 + br + c) = 0\), so \(r\) must solve the characteristic equation \(ar^2 + br + c = 0\). Its roots determine the form of the solution: real distinct, repeated, or complex.
ar^2 + br + c = 0Case 1: distinct real roots
If \(r_1 \ne r_2\) are real, \(y = c_1e^{r_1x} + c_2e^{r_2x}\). For \(y'' - 5y' + 6y = 0\), \(r^2 - 5r + 6 = (r - 2)(r - 3)\), so \(y = c_1e^{2x} + c_2e^{3x}\).
Case 2: a repeated root
If the root \(r\) is repeated, the second solution picks up a factor of \(x\): \(y = (c_1 + c_2x)e^{rx}\). For \(y'' + 4y' + 4y = 0\), \(r = -2\) twice, so \(y = (c_1 + c_2x)e^{-2x}\).
Case 3: complex roots
If \(r = \alpha \pm \beta i\), Euler's formula \(e^{i\theta} = \cos\theta + i\sin\theta\) gives real solutions \(y = e^{\alpha x}(c_1\cos\beta x + c_2\sin\beta x)\). For \(y'' + 2y' + 5y = 0\), \(r = -1 \pm 2i\), so the solution oscillates with frequency 2 while decaying like \(e^{-x}\).
y = e^{\alpha x}(c_1\cos\beta x + c_2\sin\beta x)Applying initial conditions
Solve \(y'' - y = 0\), \(y(0) = 2\), \(y'(0) = 0\). Roots \(\pm 1\) give \(y = c_1e^x + c_2e^{-x}\). Then \(c_1 + c_2 = 2\) and \(c_1 - c_2 = 0\), so \(c_1 = c_2 = 1\) and \(y = e^x + e^{-x} = 2\cosh x\).
Linear independence and the Wronskian
Two solutions are independent if neither is a constant multiple of the other. The Wronskian \(W = y_1y_2' - y_1'y_2\) is nonzero for independent solutions. For \(e^{2x}\) and \(e^{3x}\), \(W = e^{5x} \ne 0\). Independence guarantees that every solution can be written as \(c_1y_1 + c_2y_2\).
W = y_1 y_2' - y_1' y_2Superposition
For homogeneous linear equations, any linear combination of solutions is a solution. This principle fails for nonlinear equations, and it is the deep reason linear equations are so much easier to solve and analyse.
Key takeaways
Substitute \(e^{rx}\) to get the characteristic equation. Distinct real roots give exponentials; a repeated root adds a factor of \(x\); complex roots give exponentially scaled sines and cosines. Two independent solutions (nonzero Wronskian) and two initial conditions determine the unique solution.
Flashcard
Homogeneous constant-coefficient equation
Check your understanding
The characteristic equation of y'' + 9y = 0 has roots plus and minus 3i. The general solution is: