Differential EquationsDifferential EquationsCollege

Second-Order Linear (Constant Coefficients)

Characteristic equation with real, repeated or complex roots.

A second-order linear equation with constant coefficients, $ay'' + by' + cy = 0$ , describes springs, pendulums and electrical circuits. Trying $y = e^{rx}$ turns it into the characteristic equation $ar^2 + br + c = 0$ , an ordinary quadratic whose roots determine the shape of every solution.

Two distinct real roots give a sum of two exponentials; a repeated root $r$ gives $(C_1 + C_2 x)e^{rx}$ ; complex roots $\alpha \pm \beta i$ give oscillations $e^{\alpha x}(C_1\cos\beta x + C_2\sin\beta x)$ . Two initial conditions, usually $y(0)$ and $y'(0)$ , fix the two constants.

Key terms

Characteristic equation: Substituting y = e^{rx} gives a quadratic in r.
$ay'' + by' + cy = 0 \Rightarrow ar^2 + br + c = 0$

Key ideas

Characteristic equation: replace $y''$ with $r^2$ , $y'$ with $r$ , $y$ with 1.
Real distinct roots: $y = C_1 e^{r_1 x} + C_2 e^{r_2 x}$ .
Complex roots produce sines and cosines; negative real part means damping.
Non-homogeneous equations add a particular solution to this general solution.

Worked example

Solve $y'' - 5y' + 6y = 0$ with $y(0) = 2$ , $y'(0) = 5$ .

Characteristic: $r^2 - 5r + 6 = 0$ , roots $r = 2, 3$ .
General: $y = C_1 e^{2x} + C_2 e^{3x}$ .
$y(0)$ : $C_1 + C_2 = 2$ ; $y'(0)$ : $2C_1 + 3C_2 = 5$ .
So $C_2 = 1$ , $C_1 = 1$ : $y = e^{2x} + e^{3x}$ .

Answer: $y = e^{2x} + e^{3x}$

Common mistakes

Using $xe^{rx}$ when the roots are distinct.
Forgetting the $x$ factor when the root is repeated.
Applying both initial conditions to $y$ instead of differentiating for $y'(0)$ .

Practice questions

Try each one, then open the answer. Numerical answers on this page are verified by an automated calculation check.

1.What is the sum of the roots of the characteristic equation of $y'' + 4y' + 3y = 0$ ?
Show answer
$-4$
2.For $y'' + 9y = 0$ , what is the angular frequency $\beta$ ?
Show answer
$3$
3.For $y = e^{2x} + e^{3x}$ , find $y'(0)$ .
Show answer
$5$

Key terms

Key ideas

Worked example

Common mistakes

Practice questions

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