Undetermined Coefficients and Variation of Parameters: every key term you need (+ practice quiz)
25 flashcard terms for Differential Equations Topic 5, written to match the course framework. Study them here, then drill them as interactive flashcards, or test yourself with the 15-question quiz — free, no account needed.
The forcing function on the right side of a linear equation, representing an external input to the modelled system. Its shape decides which particular-solution technique is efficient.
Method of undetermined coefficients
A technique that guesses a particular solution of the same functional family as the forcing term, with unknown constants fixed by substitution. It is fast but only applies to a restricted class of forcing functions.
Admissible forcing families
Polynomials, exponentials, sines and cosines, and finite products of these. What they share is that repeated differentiation keeps producing terms from the same finite family, which is why a finite guess can work.
Polynomial forcing guess
For a polynomial forcing term of degree n, try a general polynomial of the same degree with every coefficient unknown, not just the leading one. Lower terms are needed because differentiation mixes degrees.
Exponential forcing guess
For an exponential forcing term, try a constant multiple of the same exponential. Substituting evaluates the characteristic polynomial at the exponent, so the guess fails exactly when that exponent is a characteristic root.
Sinusoidal forcing guess
For a cosine or sine forcing term, try a combination of both a cosine and a sine at the same frequency. Including only the matching function fails because differentiation converts one into the other.
Resonant guess modification
When the trial form already solves the homogeneous equation, multiply it by the independent variable, repeating once for each level of duplication. Skipping this step produces an unsolvable system of coefficient equations.
Product forcing guess
For a product such as an exponential multiplying a polynomial multiplying a sinusoid, take the product of the individual trial forms with a full set of unknown coefficients. The guess grows quickly but stays finite.
Superposition of forcing terms
If the forcing is a sum of admissible pieces, find a particular solution for each piece separately against the same left-hand side and add them. Linearity guarantees the sum solves the combined problem.
Annihilator approach
A systematic version of the guessing method in which a differential operator killing the forcing term is applied to the whole equation. The homogeneous solution of the enlarged operator supplies the correct trial form automatically.
Variation of parameters
A general technique that replaces the arbitrary constants of the complementary solution by unknown functions, then imposes a convenient side condition to obtain a solvable algebraic system for their derivatives.
Imposed side condition in variation of parameters
The requirement that the derivatives of the unknown coefficient functions contribute nothing to the first derivative. It is a free choice that removes second derivatives of the unknowns and makes the resulting system square.
Variation of parameters formula
The unknown coefficient derivatives are ratios of small determinants over the Wronskian, and the particular solution is obtained by integrating them and recombining with the fundamental solutions.
Normalisation before variation of parameters
The equation must be divided through so the second derivative has coefficient one before the forcing term is inserted into the formula. Forgetting this scales the answer by the missing coefficient.
When to prefer variation of parameters
Whenever the forcing is not in the admissible family, such as a tangent, a secant, a logarithm or a general quotient, and whenever the coefficients are not constant but a fundamental set is already known.
Cost of variation of parameters
It always works in principle but requires two integrals that may be impossible in closed form. Undetermined coefficients requires only algebra, which is why it is preferred when it applies.
Green function for an initial value problem
A kernel built from the fundamental solutions and their Wronskian such that integrating it against the forcing term gives the particular solution with zero initial data. It packages variation of parameters into one formula.
Steady state response
The part of a forced solution that persists after the transient decays, having the same form as the forcing for periodic input. For a stable equation it is independent of the initial conditions.
Transient response of a forced system
The complementary part of a forced solution, determined by the initial data and decaying when all characteristic roots have negative real part. Its decay rate fixes how long the system takes to settle.
Amplitude of the sinusoidal response
The magnitude of the steady periodic output, obtained by combining the cosine and sine coefficients of the particular solution as the square root of the sum of their squares.
Phase shift of the response
The lag of the steady output behind the sinusoidal input, recovered from the ratio of the sine and cosine coefficients through an inverse tangent with attention to the correct quadrant.
Duplication counting
The number of times the trial form appears in the complementary solution, which equals the multiplicity of the corresponding characteristic root. It fixes the power of the independent variable in the corrected guess.
Checking a particular solution
Substituting the completed candidate back into the original equation and confirming the identity term by term. This catches arithmetic errors and missing resonant factors before initial conditions are applied.
Order of operations for forced problems
Solve the homogeneous equation first, then build the particular solution, then add them, and only then impose the initial conditions on the combined general solution. Fitting constants too early is the most common error.
Nonconstant coefficient forced equations
Undetermined coefficients relies on constant coefficients and does not extend, so with variable coefficients you must first obtain a fundamental set by other means and then apply variation of parameters.