Cauchy-Euler equations and variable coefficients
Not every equation has constant coefficients. Cauchy-Euler equations are an important variable-coefficient family that can still be solved exactly.
The Cauchy-Euler form
A Cauchy-Euler (equidimensional) equation has the form \(ax^2y'' + bxy' + cy = 0\): each derivative is multiplied by the matching power of \(x\). They arise in problems with radial symmetry, such as heat flow in a disk or stress in a thick pipe.
Trying a power
Substitute \(y = x^m\). Since \(x^2y'' = m(m-1)x^m\) and \(xy' = mx^m\), the equation becomes \(am(m-1) + bm + c = 0\), the indicial equation. Its roots determine the solutions, mirroring the constant-coefficient case.
am(m-1) + bm + c = 0The three cases
Distinct real roots \(m_1, m_2\): \(y = c_1x^{m_1} + c_2x^{m_2}\). A repeated root \(m\): \(y = (c_1 + c_2\ln x)x^m\). Complex roots \(\alpha \pm \beta i\): \(y = x^\alpha(c_1\cos(\beta\ln x) + c_2\sin(\beta\ln x))\). The logarithm replaces the factor \(x\) from the constant-coefficient case.
Worked example
Solve \(x^2y'' - 2xy' + 2y = 0\) for \(x > 0\). The indicial equation \(m(m-1) - 2m + 2 = m^2 - 3m + 2 = 0\) has roots 1 and 2. So \(y = c_1x + c_2x^2\). Check: \(y = x^2\) gives \(2x^2 - 4x^2 + 2x^2 = 0\).
Why logarithms appear
The substitution \(x = e^t\) turns a Cauchy-Euler equation into a constant-coefficient equation in \(t\). Solutions \(e^{mt}\) become \(x^m\), and \(te^{mt}\) becomes \(x^m\ln x\). This change of variables explains every case at once.
Variable coefficients in general
Most equations with variable coefficients, such as Airy's equation \(y'' - xy = 0\) or Bessel's equation, have no solutions in elementary functions. Series methods, covered in the next unit, and numerical methods handle them. Special functions like Bessel functions are named solutions of such equations.
The Wronskian and Abel's theorem
For \(y'' + p(x)y' + q(x)y = 0\), Abel's theorem gives the Wronskian of any two solutions as \(W = Ce^{-\int p\,dx}\), so it is either always zero or never zero on an interval. This is a quick test of independence that does not require solving the equation.
Common mistakes
Applying constant-coefficient methods to Cauchy-Euler equations. Forgetting the restriction \(x > 0\) (or using \(|x|\)). Mishandling the repeated-root case, which needs \(\ln x\), not \(x\).
Key takeaways
Cauchy-Euler equations \(ax^2y'' + bxy' + cy = 0\) are solved with \(y = x^m\) and the indicial equation. Repeated roots bring \(\ln x\), and complex roots bring \(\cos(\beta\ln x)\) and \(\sin(\beta\ln x)\). The substitution \(x = e^t\) explains these forms. Most other variable-coefficient equations need series or numerical methods.
Flashcard
ax^2y'' + bxy' + cy = 0
Check your understanding
For x^2 y'' + x y' - 4y = 0, the indicial equation m^2 - 4 = 0 gives the solution: