Higher Order Differential Equations

What This Chapter Covers

Section	What you will learn
4.1 The Use of One Known Solution to Find Another	Given one solution $y_1$, the substitution $y_2=v(x)y_1(x)$ produces a second independent one, even for variable coefficients.
4.2 Homogeneous Equations with Constant Coefficients	The characteristic equation, and what real, repeated, and complex roots contribute.
4.3 Particular Solutions by Undetermined Coefficients	Guessing the form of $y_p$ from the shape of $f(x)$, with the Modification and Superposition Rules.
4.4 Particular Solutions by Variation of Parameters	A universal method that works for any $f(x)$ and any linear equation whose homogeneous solutions are known.
4.5 The Cauchy-Euler Equation	The one important variable-coefficient family that reduces to constant coefficients by a substitution.

The Overall Strategy

That last point deserves emphasis: apply initial conditions last, to the complete solution $y_h+y_p$, never to $y_h$ alone. Fitting the constants before adding $y_p$ is the most common procedural error in this material.

For half one, the method depends on the coefficients:

• Constant coefficients: factor the characteristic polynomial. Section 4.2.
• Cauchy-Euler form $x^n{y}^{(n)}+\cdots$: substitute $y=x^r$ or $x=e^t$. Section 4.5.
• One solution already known: reduce the order with $y=v(x)y_1(x)$. Section 4.1.
• Anything else: power series, covered in Chapter 6.

For half two, the choice is between two methods:

• Undetermined coefficients is faster, but requires constant coefficients and a forcing function built from polynomials, exponentials, sines, and cosines.
• Variation of parameters is slower but universal: any coefficients, any forcing function.

Where Second-Order Equations Appear

Second-order linear equations with constant coefficients describe an enormous range of physical systems, and remarkably they are all the same equation wearing different labels.

• Mass on a spring. $m\ddot{x}+c\dot{x}+kx=F(t)$, with $m$ the mass, $c$ the damping, and $k$ the spring constant.
• Series RLC circuit. $L\ddot{q}+R\dot{q}+{\displaystyle \frac{q}{C}}=E(t)$, identical in form, with charge in place of displacement.
• Pendulum, small oscillations. $\ddot{\theta }+{\displaystyle \frac{g}{L}}\theta =0$, the linearization of the nonlinear pendulum equation.
• Beam deflection. A fourth-order equation $EI{y}^{(4)}=w(x)$ for a loaded beam.

The three cases of the characteristic equation correspond exactly to the three physical regimes of a damped oscillator: distinct real roots give overdamping, a repeated real root gives critical damping, and complex roots give underdamping with visible oscillation.

Frequently Asked Questions

Why do exponentials solve constant-coefficient equations?

Because $e^{rx}$ is the one function whose derivatives are all multiples of itself. A linear combination of $y$, y', and y'' can cancel to zero only if all three terms are of the same type, and the exponential is the family with that property. Substituting $y=e^{rx}$ turns the differential equation into a polynomial equation in $r$.

What happens when the characteristic equation has repeated roots?

You get fewer distinct exponentials than the order of the equation, so extra solutions must be manufactured. Multiplying by powers of $x$ does it: a double root $r_1$ contributes $e^{r_{1}x}$ and $x{e}^{r_{1}x}$, a triple root adds $x^2{e}^{r_{1}x}$, and so on. Section 4.2 gives three separate explanations for why this works.

Which method should I use for the particular solution?

Try undetermined coefficients first, since guessing a form and matching coefficients is much faster than computing integrals. Fall back to variation of parameters when the coefficients are not constant, or when $f(x)$ is something like $\tan x$, $\ln x$, or $1/x$ whose derivatives never repeat.

Do these methods work for nonlinear equations?

No. Everything here depends on superposition, which holds only for linear equations. The decomposition $y=y_h+y_p$, the characteristic equation, and both particular-solution methods all fail without linearity.