Cauchy-Euler equations

Section 2.4 Cauchy-Euler equations

Definition 2.4.1. Cauchy-Euler equations.

A second-order linear homogeneous ODE with variable coefficients is Cauchy-Euler if it is of the form

\begin{equation} at^2y''+bty'+cy=0, \quad (0,\infty),\tag{2.4.1} \end{equation}

where \(a, b, c \in \mathbb{R}\text{.}\)

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Example 2.4.2.

Consider the first-order case

\begin{equation} aty'+by=0, \quad t>0,\tag{2.4.2} \end{equation}

which is a separable ODE. Using separation of variables, we have

\begin{equation*} \displaystyle\int \dfrac{1}{y}\mathrm{d}y = \displaystyle\int\dfrac{m}{x} \mathrm{d}x, \end{equation*}

where \(m=-\dfrac{b}{a}\text{.}\) Then,

\begin{equation*} \ln(|y|)=m\ln(|t|)+k=\ln(|t^m|)+k, \end{equation*}

or \(y(t)=ct^m\text{.}\)

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Since (2.4.1) and (2.4.2) are of the same form, it is reasonable to assume that functions of the form \(y(t)=t^m\) solve (2.4.1) for \(t>0\text{.}\) As with constant coefficients, we need to know the value(s) of \(m\) such that this is the case. Substituting \(y(t)=t^m\) into (2.4.1) using the power rule, we get

\begin{equation*} at^2m(m-1)t^{m-2}+btmt^{m-1}+ct^m=0. \end{equation*}

Notice the common factor of \(t^m\) which we may divide off since \(t>0\text{,}\) giving us

\begin{equation*} am(m-1)+bm+c=0. \end{equation*}

This justifies the following definition.

Definition 2.4.3. Characteristic polynomial for second-order Cauchy-Euler equations.

Given (2.4.1), there is a corresponding polynomial called the characteristic polynomial,

\begin{equation*} am^2+(b-a)m+c=0, \end{equation*}

in the variable \(m\) whose roots determine the form of its (homogeneous) solution.

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Using the substitution \(w=\ln(t)\) and the chain rule, (2.4.1) can be written as a linear homogeneous ODE with constant coefficients:

\begin{equation*} \dfrac{\mathrm{d}y}{\mathrm{d}t}=\dfrac{\mathrm{d}w}{\mathrm{d}t}\dfrac{\mathrm{d}y}{\mathrm{d}w} = \dfrac{1}{t}\dfrac{\mathrm{d}y}{\mathrm{d}w}, \end{equation*}

which yields

\begin{equation*} \dfrac{\mathrm{d}y}{\mathrm{d}w} = t \dfrac{\mathrm{d}y}{\mathrm{d}t}. \end{equation*}

Similarly,

\begin{equation*} \dfrac{\mathrm{d}^2y}{\mathrm{d}t^2} = -\dfrac{1}{t^2} \dfrac{\mathrm{d}y}{\mathrm{d}w}+\dfrac{1}{t}\dfrac{\mathrm{d}}{\mathrm{d}t} \dfrac{\mathrm{d}y}{\mathrm{d}w}, \end{equation*}

which becomes

\begin{equation*} t^2 \dfrac{\mathrm{d}^2y}{\mathrm{d}t^2} = \dfrac{\mathrm{d}^2y}{\mathrm{d}w^2}-\dfrac{\mathrm{d}y}{\mathrm{d}w}. \end{equation*}

Upon substitution, (2.4.1) becomes

\begin{equation*} ay''+(b-a)y'+cy=0. \end{equation*}

We will take advantage of this relationship in later sections. Cauchy-Euler equations are also useful in solving more general linear equations with variable coefficients by power series.

Remark 2.4.4.

Using the subtitution above, the roots of the characertistic polynomial lead to the following forms of the general solution to (2.4.1):

Table 2.4.5. General solutions to (2.4.1)

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Roots	General solution
Two real, distinct roots \(m_1\neq m_2\)	\(y(t)=c_1t^{m_1}+c_2t^{m_2}\)
One real, repeated root \(m_1=-\dfrac{b-a}{2a}\)	\(y(t)=c_1t^{m_1}+c_2 t^{m_1}\ln(t)\)
Complex roots \(r=\alpha \pm \beta i\)	\(y(t)=t^{\alpha}\Big[ c_1\cos(\beta \ln(t))+c_2\sin(\beta\ln(t)) \Big]\)

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Example 2.4.6.

Consider the differential equation \(t^2y''+2ty'-12y=0, t>0\text{.}\)

Classify the differential equation by order, linearity, type of coefficients, and whether or not the equation is homogeneous or nonhomogeneous.

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Solve the differential equation.

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Example 2.4.7.

Consider the differential equation \(t^2y''+11ty'+25y=0, t>0\text{.}\)

Classify the differential equation by order, linearity, type of coefficients, and whether or not the equation is homogeneous or nonhomogeneous.

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Solve the differential equation.

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Example 2.4.8.

Consider the differential equation \(t^2y''-3ty'+20y=0, t>0\text{.}\)

Classify the differential equation by order, linearity, type of coefficients, and whether or not the equation is homogeneous or nonhomogeneous.

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Solve the differential equation.

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