PHYS2150 Chap.8 Linear Higher-Order Differential Equations
Linear Higher-Order Differential Equations
Order determines how many solutions are needed
A linear nth-order ordinary differential equation has a homogeneous solution space with n independent directions on an interval where the normalized coefficients are continuous. That counting rule is a practical check: the complementary solution must contain n independent arbitrary constants before initial data are applied.
The Wronskian tests a proposed set for linear independence. If it is nonzero at an appropriate point for solutions of the same homogeneous equation, the functions form a fundamental set there.
Existence and uniqueness statements are interval statements, so singular leading coefficients must be identified before claiming that a solution extends globally.
Characteristic roots build the complementary solution
For constant coefficients, substituting an exponential e^{rx} changes differentiation into multiplication by powers of r and produces the characteristic polynomial.
Distinct real roots generate distinct exponentials. A real root repeated m times requires e^{rx}, xe^{rx}, through x^{m-1}e^{rx}; omitting those powers leaves too few independent functions. A complex conjugate pair α±iβ is written in real form as e^{αx}cos βx and e^{αx}sin βx. The real part controls exponential growth or decay, while the imaginary part sets oscillation.
Root multiplicity and conjugate pairing should be recorded explicitly before constants are introduced.
Nonhomogeneous problems split by superposition
The complete solution is y_c+y_p: a complementary part satisfying the homogeneous equation and one particular solution responding to the forcing. For a sum of forcing terms, linearity allows separate particular solutions that are added afterward.
Undetermined coefficients works when derivatives keep the forcing inside a finite family, including polynomials, exponentials, sines, cosines and their products. A polynomial trial includes every lower power; a trigonometric trial includes both sine and cosine; product forcing combines these families.
Coefficients stay symbolic until substitution into the original equation produces an algebraic system.
Resonance changes the trial
Before solving coefficients, compare the ordinary trial with the complementary basis. If it overlaps the homogeneous kernel, multiply the entire trial by x enough times to restore independence.
The required power equals the multiplicity of the corresponding forcing root in the characteristic polynomial. This modification applies to all companion terms, not only the visibly duplicated one, because differentiation mixes the family. Substituting an unmodified resonant trial leads to inconsistency or zero response.
Showing the overlap and the chosen multiplier makes the method auditable and addresses a known source of errors in this course’s historical performance feedback.
Methods and physics provide checks
Variation of parameters handles forcing beyond the undetermined-coefficients families. It starts from a verified fundamental set and its Wronskian after the equation has been normalized.
Constants inside the parameter integrals may be omitted because they only reproduce complementary terms. Regardless of method, verify y_p alone in the nonhomogeneous equation, then add y_c and impose all initial conditions on the complete solution. In physical systems, complementary terms describe transients determined by initial state, while the particular term follows sustained forcing.
Stable transients decay, positive real roots grow, and imaginary parts oscillate. Long-time behaviour, dimensions and the required number of initial conditions offer independent checks on the algebra.
What this chapter covers
- 01
Characteristic equations
- 02
Repeated and complex roots
- 03
Wronskians and independence
- 04
Undetermined coefficients
- 05
Variation of parameters and resonance
Worked example · free
Repair a resonant trial
- 3Find the complementary basis.
- 2Compare the forcing family.
- 3Apply the correct power multiplier.
Key terms
- Characteristic Equation
- The algebraic polynomial equation obtained by substituting e^{rx} into a homogeneous constant-coefficient ODE; its roots and multiplicities determine a complementary basis.
- Resonance Modification
- Multiplication of the complete particular trial by enough powers of the independent variable to remove overlap with the complementary solution and restore independence.
- Wronskian
- A determinant formed from functions and successive derivatives that tests whether a proposed collection of homogeneous solutions is linearly independent on an interval.
- Complementary Solution
- The general solution of the associated homogeneous linear equation, containing as many independent constants as the differential equation’s order.
- Particular Solution
- Any one solution of the nonhomogeneous equation; adding it to the complementary solution produces the full general solution.
Linear Higher-Order Differential Equations FAQ
What does a repeated characteristic root change?
A real root with multiplicity m contributes m functions: e^{rx}, xe^{rx}, and successive powers through x^{m-1}e^{rx}. The powers create the missing independent directions. Without them, the complementary solution has fewer arbitrary constants than the equation’s order and cannot generally satisfy the required initial data.
Why include sine and cosine together in a trial?
Differentiation turns sine into cosine and cosine into negative sine, so either function generates the other during substitution. A closed trial therefore includes both companions even when only one appears in the forcing. If polynomial factors are present, each companion receives a complete polynomial of the required degree before any resonance multiplier is applied.
How are transients separated from forced response?
Complementary terms contain the arbitrary constants fixed by initial displacement, velocity or higher derivatives, and they describe the system’s transient memory. A particular term follows continuing forcing. In a stable damped model the transient terms decay, leaving the forced response, so the long-time limit checks root signs and the selected particular family.
Exam move
Separate each exercise into a structure pass and an algebra pass. First normalize the equation, record its order, form the characteristic polynomial, list roots with multiplicities, and count the complementary basis functions. Next identify the forcing family, write its full differentiation-closed trial, compare it term by term with the complementary solution, and apply one resonance multiplier to the whole family.
Only then substitute for coefficients. Practise a second set where you stop after trial selection, because this isolates the decision students most often lose marks on. End full solutions by verifying the particular term, applying every initial condition to the combined solution, and describing transient versus long-time behaviour.
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