Home » Blog » Prove complex valued functions of a given form satisfy a differential equation

Prove complex valued functions of a given form satisfy a differential equation

by RoRi

Let L be defined by

    \[ L(y) = y'' + ay' + by, \]

for a, b \in \mathbb{R} constants.

Let R be a complex-valued function so that R(x) = P(x) + i Q(x) for functions P(x) and Q(x). Prove that a function

    \[ f(x) = u(x) + i v(x) \]

satisfies the differential equation

    \[ L(y) = R(x) \qquad x \in I \]

if and only if u and v satisfy the differential equations

    \[ L(u) = P(x) \qquad L(v) = Q(x) \qquad x \in I. \]

Proof. (\Rightarrow) Assume y = f(x) = u(x) + i v(x) satisfies the differential equation L(y) = R(x) for all x \in I. Then,

    \begin{align*} L(y) = R(x) && \implies && f''(x) + af'(x) + bf(x) &= P(x) + i Q(x) \\[9pt] && \implies && u''(x) + iv''(x) + au'(x) + iav'(x) + bu(x) + bv(x) &= P(x) + i Q(x). \end{align*}

Setting real and imaginary parts equal we have

    \[ u''(x) + au'(x) + bu(x) = P(x) \quad \text{and} \quad v''(x) + av'(x) + bv(x) = Q(x). \]

Thus,

    \[ L(u) = P(x) \qquad \text{and} \qquad L(v) = Q(x) \qquad \text{for all } x \in I. \qquad \blacksquare\]

(\Leftarrow) Assume L(u) = P(x) and L(v) = Q(x) for all x \in I. Then we have

    \[ u''(x) + au'(x) + bu(x) = P(x) \qquad \text{and} \qquad v''(x) + av'(x) + bv(x) = Q(x). \]

Therefore,

    \begin{align*} u''(x) + au'(x) + bu(x) + i \left( v''(x) + av'(x) + bv(x) \right) &= P(x) + iQ(x) \\ \implies \qquad f''(x) + af'(x) + bf(x) &= R(x) \\ \implies L(y) &= R(x) \qquad \text{for all } x \in I. \qquad \blacksquare \end{align*}

Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment):