6.1 The homogeneous equation
Lemma 6.1.1 (Exponential shift). For any polynomial \(q\) and smooth \(u\), \(q(D)\bigl (e^{\lambda t}u\bigr ) = e^{\lambda t}q(D + \lambda )u\). In particular \(p(D)e^{\lambda t} = p(\lambda )e^{\lambda t}\).
Proof. \(D(e^{\lambda t}u) = e^{\lambda t}(Du + \lambda u) = e^{\lambda t}(D + \lambda )u\); apply this repeatedly for powers of \(D\) and use linearity. □
Theorem 6.1.2. Let \(p(\lambda ) = \prod _j(\lambda - \lambda _j)^{m_j}\) with distinct roots \(\lambda _j\). The \(n\) functions \(t^ke^{\lambda _jt}\), \(0 \leq k < m_j\), form a basis of the solutions of \(p(D)y = 0\). For a complex pair \(\lambda = a \pm ib\) of multiplicity \(m\), the functions \(t^ke^{\lambda t}\) and \(t^ke^{\bar \lambda t}\) may be replaced by the real functions \(t^ke^{at}\cos bt\) and \(t^ke^{at}\sin bt\).
Proof. Write \(p(D) = q(D)(D - \lambda _j)^{m_j}\). By Lemma 6.1.1, \((D - \lambda _j)^{m_j}(t^ke^{\lambda _jt}) = e^{\lambda _jt}D^{m_j}t^k = 0\) for \(k < m_j\), and the factors of \(p(D)\) commute, so each \(t^ke^{\lambda _jt}\) is a solution. There are \(\sum _jm_j = n\) of them, and by Theorem 4.1.1 it suffices to show they are linearly independent. Suppose \(\sum _jq_j(t)e^{\lambda _jt} = 0\) with polynomials \(q_j\) of degree less than \(m_j\), not all zero. Multiply by \(e^{-\lambda _1t}\) and differentiate \(m_1\) times: the term \(q_1\) disappears, and each other term becomes \(r_j(t)e^{(\lambda _j - \lambda _1)t}\) with \(r_j = (D + \lambda _j - \lambda _1)^{m_1}q_j\), a polynomial of the same degree as \(q_j\), since \(\lambda _j \neq \lambda _1\). This gives a relation with one fewer exponential, and induction on the number of distinct roots reduces to a single term \(r(t)e^{\mu t} = 0\) with \(r \neq 0\), which is impossible. Finally, if \(p\) has real coefficients, the real and imaginary parts of a complex solution are real solutions, and they span the same space as the conjugate pair. □
The theorem is Theorem 5.2.1(iii) for the companion matrix, whose eigenvalues are the roots of \(p\) and whose Jordan blocks have sizes equal to the multiplicities.
Euler equations
The equidimensional equation \(t^ny^{(n)} + b_{n-1}t^{n-1}y^{(n-1)} + \dots + b_0y = 0\) on \(t > 0\) becomes a constant-coefficient equation under \(t = e^s\), because \(t\frac {d}{dt} = \frac {d}{ds}\). In practice one tries \(y = t^r\): for \(n = 2\), \(t^2y'' + bty' + cy = 0\) gives \(r(r - 1) + br + c = 0\), a double root \(r\) gives the second solution \(t^r\log t\), and a complex pair \(\alpha \pm i\beta \) gives \(t^\alpha \cos (\beta \log t)\) and \(t^\alpha \sin (\beta \log t)\). Euler equations are the model for the regular singular points of the section on series solutions.
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