More methods for linear equations

Lesson 3 solved linear equations with constant coefficients and simple forcing terms: exponentials, polynomials, sines and cosines. Engineering problems often go beyond that. The forcing may be an arbitrary measured signal, such as a ground acceleration record or a load that ramps up over time. The coefficients may vary, as they do for stresses in a thick pipe wall, where the equation contains the radius. And the equation may be of fourth order, as it is for beams. This lesson adds the tools for these cases: the theory of fundamental solutions and the Wronskian, reduction of order, variation of parameters (which handles any forcing and leads to Duhamel's integral), Cauchy-Euler equations with variable coefficients, and higher-order equations, applied to a rail resting on its ballast.

Fundamental solutions and the Wronskian

For the homogeneous equation \(y'' + p(t)y' + q(t)y = 0\), Lesson 3 combined two solutions into \(y = C_1y_1 + C_2y_2\). This works only if the two solutions are genuinely different, so that any initial position and velocity can be matched. The test is the Wronskian:

If \(W \ne 0\), the two solutions form a fundamental set and \(C_1y_1 + C_2y_2\) is the general solution. If \(W = 0\) everywhere, one solution is a multiple of the other and a second, independent solution is still needed. The Wronskian is the determinant of the 2 × 2 system for \(C_1\) and \(C_2\) at the initial time, so it is exactly the condition that this system has a unique solution (MATH-125).

Reduction of order

If one solution \(y_1\) is known, a second can be found by trying \(y = v(t)\,y_1(t)\). Substituting into the equation, the terms in \(v\) itself cancel (because \(y_1\) is a solution), leaving a first-order equation for \(v'\). This is how the extra factor \(t\) in the repeated-root case of Lesson 3 is derived, and it is the standard way to complete a solution when one solution can be guessed.

Variation of parameters

The method of undetermined coefficients (Lesson 3) needs a forcing term of a special form. Variation of parameters works for any continuous forcing. For \(y'' + p(t)y' + q(t)y = g(t)\) with a known fundamental set \(y_1, y_2\), look for a particular solution \(y_p = u_1y_1 + u_2y_2\), letting the constants vary. Imposing the convenient condition \(u_1'y_1 + u_2'y_2 = 0\) and substituting gives two linear equations for \(u_1'\) and \(u_2'\), whose solution is:

Integrating these gives the particular solution:

The equation must first be divided by the coefficient of \(y''\) so that the leading coefficient is 1; forgetting this is the most common error.

Duhamel's integral

Applying variation of parameters to an undamped oscillator \(m\ddot{x} + kx = f(t)\), starting from rest, gives a formula for the response to any force:

This is Duhamel's integral. It has a clear physical meaning: the force during each short interval \(d\tau\) delivers a small impulse \(f(\tau)\,d\tau\), which sets the mass vibrating as \(\sin\omega_n(t - \tau)\) from that moment on, and the response is the sum of all these vibrations. Structural engineers evaluate it numerically for measured earthquake records, and Lesson 8 shows that it is a convolution, the same operation that Laplace transforms turn into a product.

Cauchy-Euler equations

Most linear equations with variable coefficients have no solution in elementary functions, but one important family does. A Cauchy-Euler (equidimensional) equation has the form:

Each derivative is multiplied by the matching power of \(x\), so the trial \(y = x^m\) turns every term into a multiple of \(x^m\). Substituting gives the indicial equation:

Its roots decide the form of the solution, in close analogy with the characteristic equation:

• Two real roots: \(y = C_1x^{m_1} + C_2x^{m_2}\).
• A repeated root: \(y = (C_1 + C_2\ln x)\,x^m\), found by reduction of order.
• Complex roots \(\alpha \pm i\beta\): \(y = x^{\alpha}\left[C_1\cos(\beta\ln x) + C_2\sin(\beta\ln x)\right]\).

These equations appear wherever a problem has no natural length scale, especially in polar and cylindrical coordinates: stresses in pipes, discs and around holes, and steady heat conduction through cylindrical walls.

Higher-order equations

Everything in Lesson 3 extends to linear constant-coefficient equations of any order. For an nth-order equation, the characteristic polynomial has n roots, each contributing a solution (with extra factors of \(t\) for repeated roots), and n initial or boundary conditions fix the n constants. The most important engineering example is fourth order: the Euler-Bernoulli beam equation \(EI\,y'''' = w(x)\), which relates the deflection of a beam to the load per unit length on it.

Which method to use

For linear equations, the choice now runs as follows. With constant coefficients and simple forcing, use the characteristic equation and undetermined coefficients (Lesson 3), or Laplace transforms (Lessons 7 and 8) for switched and impulsive inputs. For arbitrary forcing, use variation of parameters or its integral form, Duhamel's integral. For variable coefficients of the Cauchy-Euler type, try powers of \(x\); for other variable coefficients, use series solutions (covered in more advanced courses) or a numerical method (MATH-300). In every case, the general solution is the homogeneous solution plus one particular solution, and the constants are fitted last.