Higher-order linear equations
The ideas for second-order equations extend naturally to third, fourth, and higher orders, which model beams, control systems, and more.
Order n linear equations
An \(n\)th-order linear equation has the form \(a_ny^{(n)} + \cdots + a_1y' + a_0y = g(x)\). Its homogeneous general solution combines \(n\) independent solutions with \(n\) constants, fixed by \(n\) initial conditions on \(y, y', \ldots, y^{(n-1)}\).
The characteristic polynomial
For constant coefficients, \(y = e^{rx}\) leads to the characteristic polynomial \(a_nr^n + \cdots + a_0 = 0\), which has \(n\) roots counted with multiplicity. Each root contributes solutions just as in the second-order case.
Building solutions from roots
Each distinct real root \(r\) contributes \(e^{rx}\). A root of multiplicity \(m\) contributes \(e^{rx}, xe^{rx}, \ldots, x^{m-1}e^{rx}\). Each complex pair \(\alpha \pm \beta i\) contributes \(e^{\alpha x}\cos\beta x\) and \(e^{\alpha x}\sin\beta x\), with extra powers of \(x\) if repeated.
Worked example: a third-order equation
Solve \(y''' - 6y'' + 11y' - 6y = 0\). The characteristic polynomial \(r^3 - 6r^2 + 11r - 6\) factors as \((r - 1)(r - 2)(r - 3)\). So \(y = c_1e^x + c_2e^{2x} + c_3e^{3x}\). Testing small integer roots, which must divide the constant term, is the usual way to begin factoring.
Worked example: repeated and complex roots
For \(y^{(4)} + 2y'' + y = 0\): \(r^4 + 2r^2 + 1 = (r^2 + 1)^2\), so \(\pm i\) each have multiplicity 2. The solution is \(y = (c_1 + c_2x)\cos x + (c_3 + c_4x)\sin x\). Repeated complex roots produce oscillations whose amplitude grows linearly.
Nonhomogeneous higher-order equations
The structure \(y = y_h + y_p\) still holds. Undetermined coefficients works the same way, including the modification rule: multiply the guess by \(x^s\), where \(s\) is the multiplicity of the forcing's exponent as a characteristic root.
Beam deflection
A uniform beam under a load \(w(x)\) bends according to \(EI\,y^{(4)} = w(x)\), where \(E\) is the material's stiffness and \(I\) depends on the beam's cross-section. Four boundary conditions, such as zero deflection and zero slope at a clamped end, determine the shape. Engineers use this to size beams in buildings and bridges.
EI\,y^{(4)} = w(x)Reduction of order
If one solution \(y_1\) of a homogeneous linear equation is known, substituting \(y = v(x)y_1(x)\) reduces the equation to a lower order in \(v'\). This explains where the factor \(x\) comes from in the repeated-root case and helps solve equations with variable coefficients.
Key takeaways
\(n\)th-order linear equations need \(n\) independent solutions and \(n\) conditions. Roots of the characteristic polynomial build the solution, with powers of \(x\) for repeated roots and sines and cosines for complex ones. Undetermined coefficients extends naturally, and fourth-order equations model beam bending.
Flashcard
a_ny^{(n)} + \cdots + a_0y = g
Check your understanding
How many arbitrary constants does the general solution of a fourth-order linear ODE have?