Higher-Order Linear Equations with Constant Coefficients

This chapter moves from first-order to $n$th-order linear equations, and specialises immediately to the case most of physics and engineering actually needs: constant coefficients. The remarkable fact proved here is that solving such an equation reduces entirely to algebra — factoring a polynomial — rather than to any new integration technique. The chapter builds the linear-algebra scaffolding (linear independence, the Wronskian, superposition) that makes precise why a handful of building-block solutions assembles into the complete general solution, then works out all three cases the characteristic polynomial's roots can take.

Have to hand: factoring and the quadratic formula, complex numbers and Euler's formula $e^{i\theta}=\cos\theta+i\sin\theta$ (used freely here; Chapter 10 revisits it rigorously), and the determinant of a $2\times2$ (or $n\times n$) matrix.

1. The General Linear Equation and Superposition

An $n$th-order linear ODE has the form

It is homogeneous (zero forcing term — the Chapter 2 sense of the word) if $g(x)\equiv 0$, and has constant coefficients if every $a_i$ is a constant. This chapter treats

Superposition principle. If $y_1,\dots,y_n$ are each solutions of a linear homogeneous equation, then so is any linear combination $c_1y_1+\cdots+c_ny_n$. This follows immediately from linearity of the differential operator $L[y]=a_ny^{(n)}+\cdots+a_0y$: $L[c_1y_1+\cdots+c_ny_n] = c_1L[y_1]+\cdots+c_nL[y_n] = 0$.

Chapter 1's existence-uniqueness theory extends to this setting: an IVP for an $n$th-order linear equation with continuous coefficients and $n$ initial conditions $y(x_0),y'(x_0),\dots,y^{(n-1)}(x_0)$ has a unique solution on the whole interval of continuity of the coefficients (no finite-time blow-up, unlike the nonlinear case). This chapter assumes that theorem and focuses on constructing the solution explicitly.

2. Linear Independence and the Wronskian

The general solution of an $n$th-order linear homogeneous equation is a linear combination of $n$ solutions — but only if those $n$ solutions are genuinely independent of one another.

Definition. Functions $y_1,\dots,y_n$ are linearly independent on an interval $I$ if the only constants $c_1,\dots,c_n$ satisfying $c_1y_1(x)+\cdots+c_ny_n(x)=0$ for every $x\in I$ are $c_1=\cdots=c_n=0$.

The practical test uses the Wronskian

Theorem. If $y_1,\dots,y_n$ solve the same $n$th-order linear homogeneous equation with continuous coefficients on $I$, then either $W\equiv 0$ on $I$ (and the $y_i$ are linearly dependent) or $W(x)\ne 0$ for every $x\in I$ (and the $y_i$ are linearly independent).

Proof sketch. If the $y_i$ were linearly dependent, some nontrivial combination vanishes identically, and the same combination of their derivatives up to order $n-1$ also vanishes identically, making the columns of the Wronskian matrix linearly dependent at every $x$, so $W\equiv 0$. Conversely, one shows $W$ itself satisfies a first-order linear ODE (Abel's formula, $W'=-\dfrac{a_{n-1}}{a_n}W$), whose solution $W(x)=W(x_0)e^{-\int (a_{n-1}/a_n)dx}$ is an exponential and therefore either identically zero (if $W(x_0)=0$) or never zero. $\blacksquare$

This theorem is why checking $n$ solutions' Wronskian at a single convenient point (often $x=0$) is enough to certify independence everywhere.

3. The Characteristic Equation

Seek solutions of $a_ny^{(n)}+\cdots+a_0y=0$ of the trial form $y=e^{rx}$. Since $y^{(k)}=r^ke^{rx}$, substituting gives

As $e^{rx}\ne 0$, this holds exactly when $r$ is a root of the characteristic (auxiliary) polynomial

So: every root $r$ of this algebraic equation gives a solution $y=e^{rx}$ of the differential equation — the entire problem of solving the ODE has been converted into the entirely algebraic problem of factoring a polynomial. By the fundamental theorem of algebra, the characteristic polynomial has exactly $n$ roots counted with multiplicity (possibly complex, possibly repeated), and the three cases below cover every possibility.

4. Case 1: Distinct Real Roots

If $r_1,\dots,r_n$ are real and all different, the $n$ functions $e^{r_1x},\dots,e^{r_nx}$ are linearly independent (their Wronskian is a nonzero multiple of the Vandermonde determinant in the $r_i$, which is nonzero exactly because the $r_i$ are distinct), so the general solution is

Worked example. Solve $y''-y'-6y=0$. Characteristic equation: $r^2-r-6=0$, factoring as $(r-3)(r+2)=0$, so $r=3,-2$. General solution:

5. Case 2: Repeated Real Roots

If $r$ is a root of multiplicity $k$, the trial form $e^{rx}$ only yields one independent solution where $k$ are needed. Reduction of order supplies the rest: given one solution $y_1=e^{rx}$ of a second-order equation, seek a second solution of the form $y_2=v(x)e^{rx}$ for an unknown function $v$. Substituting into $y''+py'+qy=0$ (with $p,q$ constants here) and using that $r$ is a repeated root of $r^2+pr+q=0$ (so $r=-p/2$ and the first-derivative terms cancel), one finds $v''=0$, i.e. $v(x)=Ax+B$. Taking $v=x$ gives the second independent solution $xe^{rx}$.

Pattern for multiplicity $k$. A root $r$ repeated $k$ times contributes exactly the $k$ independent solutions

$$e^{rx},\ xe^{rx},\ x^2e^{rx},\ \ldots,\ x^{k-1}e^{rx}.$$

Worked example. Solve $y''-6y'+9y=0$. Characteristic equation: $r^2-6r+9=(r-3)^2=0$, so $r=3$ with multiplicity $2$. General solution:

6. Case 3: Complex Conjugate Roots

Since the characteristic polynomial has real coefficients, complex roots occur in conjugate pairs $r=\alpha\pm i\beta$ ($\beta\ne 0$). The trial solutions $e^{(\alpha+i\beta)x}$ and $e^{(\alpha-i\beta)x}$ are complex-valued; by Euler's formula,

Because the equation is linear with real coefficients, the real and imaginary parts of any complex solution are themselves real solutions (if $L[u+iv]=0$ with $u,v$ real-valued and $L$ has real coefficients, then $L[u]+iL[v]=0$, and since $L[u],L[v]$ are both real, each must vanish separately). Taking real and imaginary parts of $e^{(\alpha+i\beta)x}$ gives the two independent real solutions

A complex conjugate pair repeated $k$ times contributes $2k$ real solutions by multiplying this pattern by $1,x,\dots,x^{k-1}$, exactly as in Case 2.

Worked example. Solve $y''+2y'+5y=0$. Characteristic equation: $r^2+2r+5=0$, giving $r=\dfrac{-2\pm\sqrt{4-20}}{2} = -1\pm 2i$, so $\alpha=-1,\beta=2$. General solution:

7. Assembling the General Solution for Mixed Cases

A characteristic polynomial of degree $n$ in general has some distinct real roots, some repeated real roots, and some complex conjugate pairs (possibly repeated); the general solution is the sum of the contribution from each case, with $n$ arbitrary constants in total — guaranteed linearly independent as a full set by an extension of the Wronskian argument of Section 2.

Worked example (fourth order, mixed). Solve $y^{(4)} - y = 0$. Characteristic equation: $r^4-1=0$, factoring as $(r^2-1)(r^2+1)=0$, so $r=1,-1$ (real, distinct) and $r=\pm i$ (complex, $\alpha=0,\beta=1$). General solution:

8. Common Exam Pitfalls

• Forgetting the extra factor of $x$ (or $x^2,\dots$) for repeated roots. Writing $y=c_1e^{3x}+c_2e^{3x}$ for a double root $r=3$ is wrong — the second term collapses into the first (they are linearly dependent), leaving only one arbitrary constant where two are needed; it must be $c_1e^{3x}+c_2xe^{3x}$.
• Writing complex exponentials instead of the real sine/cosine form. $y=c_1e^{(\alpha+i\beta)x}+c_2e^{(\alpha-i\beta)x}$ is a valid complex general solution, but for a real-valued IVP the expected answer is the real form $e^{\alpha x}(c_1\cos\beta x+c_2\sin\beta x)$ (with new, real constants related to but not identical to the complex $c_1,c_2$).
• Misreading the characteristic equation's sign convention. $y''+py'+qy=0$ gives $r^2+pr+q=0$ — the signs of $p,q$ carry over unchanged; a common error is flipping a sign when the original equation is written with a term moved to the other side.
• Using the Wronskian test on functions that do not solve the same equation. The dichotomy "$W\equiv0$ or never zero" is a theorem about solutions of a common linear ODE; for arbitrary functions, $W$ can vanish at isolated points without implying global dependence or independence.
• Dropping a root during factoring of a higher-degree characteristic polynomial. Always verify the number of roots found (with multiplicity) equals the degree $n$ of the polynomial — a degree-4 polynomial factored into only three roots signals an arithmetic slip.
• Confusing "repeated root" multiplicity with the number of arbitrary constants needed overall. Every individual root (real or complex, simple or repeated) contributes exactly as many basis solutions as its multiplicity; the total across all roots must equal $n$, matching the order of the equation.

Summary

• Superposition: linear combinations of solutions of a linear homogeneous equation are themselves solutions, by linearity of the differential operator $L$.
• Linear independence is tested via the Wronskian; for solutions of a common linear equation with continuous coefficients, $W$ is either identically zero or never zero (Abel's formula).
• Characteristic equation: trying $y=e^{rx}$ converts $a_ny^{(n)}+\cdots+a_0y=0$ into the polynomial equation $a_nr^n+\cdots+a_0=0$.
• Distinct real roots $r_i$ contribute $e^{r_ix}$ each; repeated real roots of multiplicity $k$ contribute $e^{rx},xe^{rx},\dots,x^{k-1}e^{rx}$ (derived via reduction of order); complex conjugate pairs $\alpha\pm i\beta$ contribute the real solutions $e^{\alpha x}\cos\beta x,\,e^{\alpha x}\sin\beta x$ (from Euler's formula and taking real/imaginary parts), each repeated similarly if the pair itself is repeated.
• The general solution sums the contributions of all roots, with exactly $n$ arbitrary constants for an $n$th-order equation, fixed by $n$ initial conditions via the (assumed) existence-uniqueness theorem for linear equations.