Variation of Parameters

The undetermined-coefficients method is fast but narrow: it applies only when the forcing $g (t)$ is a polynomial, exponential, sine, cosine, or a product of these, and only when the coefficients are constant. Forcing terms like $tan t$ , $sec t$ , or $1/ t$ fall outside it entirely, because their derivatives do not close into a finite family, so no finite trial form exists. Variation of parameters removes that restriction. Given any fundamental set of the homogeneous equation, it produces a particular solution of the nonhomogeneous equation for an arbitrary continuous $g$ , as an explicit formula in two integrals. Those integrals must still be evaluated, which is not always possible in closed form.

Letting the constants vary

Start from the nonhomogeneous equation and its complementary solution,

y^{''} + p (t) y^{'} + q (t) y = g (t), y_{c} = c_{1} y_{1} (t) + c_{2} y_{2} (t),

where $y_{1}, y_{2}$ are a fundamental set of the homogeneous equation. The complementary solution has fixed constants $c_{1}, c_{2}$ . The method, originally Lagrange's, promotes those constants to functions and seeks a particular solution of the form

y = u_{1} (t) y_{1} (t) + u_{2} (t) y_{2} (t) . (V)

This is one function $y$ but two unknowns $u_{1}, u_{2}$ , so one extra condition can be imposed. Differentiating (V),

y^{'} = u_{1}^{'} y_{1} + u_{2}^{'} y_{2} + u_{1} y_{1}^{'} + u_{2} y_{2}^{'} .

To keep the second derivative free of $u_{1}^{''}, u_{2}^{''}$ , impose one constraint, using the extra freedom:

u_{1}^{'} y_{1} + u_{2}^{'} y_{2} = 0. (Constraint)

With the constraint, $y^{'} = u_{1} y_{1}^{'} + u_{2} y_{2}^{'}$ , and differentiating again,

y^{''} = u_{1}^{'} y_{1}^{'} + u_{2}^{'} y_{2}^{'} + u_{1} y_{1}^{''} + u_{2} y_{2}^{''} .

Substituting $y, y^{'}, y^{''}$ into the equation and grouping, the terms multiplying $u_{1}$ and $u_{2}$ are $L [y_{1}]$ and $L [y_{2}]$ , both zero because $y_{1}, y_{2}$ solve the homogeneous equation. What remains is

u_{1}^{'} y_{1}^{'} + u_{2}^{'} y_{2}^{'} = g (t) . (Balance)

The derivation collapses a hard second-order problem into a linear two-by-two system for

u_{1}^{'}, u_{2}^{'}

by spending one imposed constraint.

Solving the system with the Wronskian

The constraint and the balance equation form a linear system in the unknowns $u_{1}^{'}, u_{2}^{'}$ :

y_{1} u_{1}^{'} + y_{2} u_{2}^{'} y_{1}^{'} u_{1}^{'} + y_{2}^{'} u_{2}^{'} = 0, = g .

Its coefficient determinant is the Wronskian $W [y_{1}, y_{2}] = y_{1} y_{2}^{'} - y_{1}^{'} y_{2}$ , which is nonzero because $y_{1}, y_{2}$ form a fundamental set. By Cramer's rule,

u_{1}^{'} (t) = - \frac{y _{2} ( t ) g ( t )}{W [ y _{1} , y _{2} ] ( t )}, u_{2}^{'} (t) = \frac{y _{1} ( t ) g ( t )}{W [ y _{1} , y _{2} ] ( t )} .

Division by $W$ is legitimate precisely because independence guarantees $W \neq = 0$ . Integrating gives $u_{1}, u_{2}$ , and (V) assembles the particular solution.

Cramer's rule on the two-by-two system delivers

u_{1}^{'}, u_{2}^{'}

as Wronskian ratios; the forcing

g

enters each numerator, the denominator is always

W

Choosing a definite lower limit of integration converts this into a closed formula for a particular solution.

The formula has two structural consequences. It is written for an arbitrary $g$ , so it directly shows how the response depends on the forcing, and with the lower limit chosen as the initial point it produces the solution satisfying $y (t_{0}) = 0$ , $y^{'} (t_{0}) = 0$ automatically.

Trigonometric-quotient forcing

A quotient of trigonometric functions cannot be handled by undetermined coefficients, because its derivatives never close into a finite family.

No finite trial form could have produced the $ln (cos t)$ term, which is why undetermined coefficients does not apply to $g = 8 tan t$ .

The tangent-forced solution lives only on the open interval

(a, b) = (- π /2, π /2)

where

tan t

is continuous; the logarithmic term drives it to large negative values as

t

nears either endpoint.

The variation-of-parameters pipeline for the worked example, from the fundamental set to the assembled particular solution.

A cosecant forcing is handled the same way, and the same fundamental set carries it through.

The response kernel

Choosing the lower limit of integration as the initial point $t_{0}$ turns the formula into the unique solution of the equation with zero initial data, $y (t_{0}) = 0$ , $y^{'} (t_{0}) = 0$ . Combining the two integrals under one sign,

Y (t) = t_{0} \int t \frac{y _{1} ( s ) y _{2} ( t ) - y _{1} ( t ) y _{2} ( s )}{W [ y _{1} , y _{2} ] ( s )} g (s) d s = t_{0} \int t K (t, s) g (s) d s .

The kernel $K (t, s)$ depends only on the homogeneous solutions, not on the forcing. For constant-coefficient equations it depends on $t$ and $s$ only through the difference $t - s$ , so $K = K (t - s)$ and the response is a convolution of the kernel with the input,

Y (t) = t_{0} \int t K (t - s) g (s) d s .

The response weights past input by the kernel

sin (t - s)

: the weight is zero at the present instant

s = t

and oscillates back over earlier times.

Reading $g$ as the input and $Y$ as the output: the output at time $t$ is a weighted sum of the input over the entire interval up to $t$ , with the kernel supplying the weights. Once $K$ is known, every forcing $g$ is handled by a single integral, and the same structure reappears in the Laplace-transform treatment of convolution, where $K$ is the inverse transform of the transfer function.

Comparison of the two methods

Variation of parameters is universal but computational; undetermined coefficients is restricted but algebraic. The choice is usually clear from the forcing term and the coefficients.

	Undetermined coefficients	Variation of parameters
Coefficients	constant only	any (given a fundamental set)
Forcing $g$	polynomial, exp, sin, cos, products	any continuous function
Work involved	solve linear equations for constants	evaluate two integrals
Resonance	handled by multiplying trial by $t$	automatic, no special case
Fails when	$g$ outside the trial families	integrals have no closed form

A practical rule: if $g$ fits the trial families and the coefficients are constant, undetermined coefficients is less work; otherwise variation of parameters is the method that applies. When both apply they give the same particular solution up to a complementary piece.

The operator viewpoint

There is also an operator formulation. Writing the equation with the differential operator $L = D^{2} + b D + c = (D - r_{1}) (D - r_{2})$ , where $r_{1}, r_{2}$ are the characteristic roots, turns solving $L [y] = g$ into inverting a product of first-order operators. Each factor $(D - r) y = h$ is a first-order linear equation solved by an integrating factor, so the second-order problem unwinds as two successive first-order integrations. This operator method recovers the same integral structure as the variation-of-parameters formula.¹

The formula extends to $n$ -th order with an $n \times n$ Wronskian, and the same determinant construction underlies nonhomogeneous linear systems through the fundamental matrix.²³

Simmons, Differential Equations with Applications and Historical Notes, §19 The Method of Variation of Parameters; §23 Operator Methods: the variation construction and the factored-operator $(D - r_{1}) (D - r_{2})$ approach to nonhomogeneous constant-coefficient equations. ↩
Boyce, Elementary Differential Equations, §3.6 — Variation of Parameters: Lagrange's substitution $y = u_{1} y_{1} + u_{2} y_{2}$ , the constraint $u_{1}^{'} y_{1} + u_{2}^{'} y_{2} = 0$ , the Wronskian formulas for $u_{1}^{'}, u_{2}^{'}$ , the general integral formula (Theorem 3.6.1), and the $y^{''} + 4 y = 8 tan t$ example. ↩
Boyce, Elementary Differential Equations, §4.4 — The Method of Variation of Parameters: the extension of the formula to $n$ -th order linear equations using the $n \times n$ Wronskian and its cofactors. ↩