This section presents a class of variable coefficient equations that admit closed form solutions---the Euler equations (also known as equidimensional equations).
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Linear differential equations with variable coefficients rarely have explicit solutions expressed via familiar functions. However, there is a special class of variable coefficient linear differential equations where explicit solutions are not difficult to obtain. They were discovered by Leonhard Euler (1707--1783).
where \( a_0 , a_1 , \ldots , a_n \) are some real constants and \( y^{(k)} (x) = {\text d}^k y/{\text d}x^k \) is the kth derivative. The general inhomogeneous Euler equation is
This equations are also known Cauchy--Euler equation or Euler--Cauchy equations. The most common term for these equations is the equidimensional equation because products of powers of x and corresponding derivatives, \( x^k y^{(k)} \ k = 0,1,2,\ldots , \) have the same units as the function y(x). This keeps the coefficients \( a_0 , a_1 , \ldots , a_n \) dimensionless. Since the leading coefficient in equation \eqref{EqEuler.2} is 0 when x = 0, this point is a singular point for Euler's equation; this forces us to consider Euler's differential equation in domains not containing the origin: either for x > 0 or x < 0. As we will see in the following examples, initial value problems with initial conditions specified at the origin are ill-posed: their solutions either do not exist or have infinite many solutions.
Euler's equations can be generalized as
\begin{equation} \label{EqEuler.3}
a_n \left( \alpha x + \beta \right)^n y^{(n)} + a_{n-1} \left( \alpha x + \beta \right)^{n-1} y^{(n-1)} + \cdots + a_1 \left( \alpha x + \beta \right) y' + a_0 y =0 \qquad (a_n \ne 0) ,
\end{equation}
with some real numbers α and β. Upon changing independent variable t = αx + β equidimensional equation \eqref{EqEuler.3} can be transferred into standard Euler's equation \eqref{EqEuler.1} having singular point at the origin instead of −β/α.
When n = 2, Euler's homogeneous equation can be written as
Thus, we get a constant coefficient differential equation for which we know everything from the previous discussion in Part IV: of this tutorial. The characteristic equation for the above constant coefficient differential equation
\begin{equation} \label{EqEuler.5}
a\,\nu^2 + \left( b - a \right) \nu + c = 0
\end{equation}
is called the indicial equation for the Euler homogeneous equation \eqref{EqEuler.3}.
Euler's equation \eqrefl{EqEuler.1} can also be solved by trial solution
y = x^{ν}. Upon substituting it into the differential equation \eqrefl{EqEuler.3}, we get
If roots ν_{1} and ν_{2} of the indicial equation are distinct real numbers, then the general solution of Euler's equation is given by
\[
y(x) = C_1 |x|^{\nu_1} + C_2 |x|^{\nu_2}
\]
for some arbitrary real constants C_{1} and C_{2}. If the roots ν_{1} and ν_{2} are equal (ν_{1} = ν_{2}), then the general solution of Euler's equation is given by
where 𝑎 and b are some real numbers. Note that both initial conditions are specified at the singular point x = 0. Substituting for y(x) = x_{m, we get an indicial equation
\[
3 \left( m^2 - m \right) -7\,m + 7 = 0 .
\]
It has two distinct real roots m = 1 and m = 7/3. Therefore, the given Euler equation has the general solution:
\[
y(x) = C_1 x^{7/3} + C_2 \, x ,
\]
where C1 and C2 are arbitrary constants.
To satisfy the initial conditions, we have to solve the system of equations
\[
C_1 0 + c_2 0 = a, \qquad C_2 = b .
\]
So 𝑎 must be zero and the given initial value problem has one-parameter set of solutions
\[
y (x) = C_1 x^{7/3} + b\, x .
\]
Note that if the arbitrary real constant C1 ≠ 0, its solution is not analytic.
■ }
Euler's equation \eqref{EqEuler.1} has a regular singular point at the origin because the leading coefficient vanishes at that point. Usually, its solution does not exist (unless the indicial equation has positive integer roots) in a neighborhood |x| < ρ (for some positive ρ) of the origin x = 0. Therefore, the initial conditions at x = 0 are not suitable for Euler's equations because existence and uniqueness theorems fail in this case. However, one condition at x = 0 can be imposed considering it as a boundary condition on some interval, say (0, ℓ), where ℓ ≤ ∞.
There is generally another singularity at the point at infinity in the complex plane ℂ, whose nature is found by mapping the point at infinity to the origin via substitution x = 1/t. The change of variable of differentiation has the effect
and the singularity at infinity is seen to be regular. For a regular singular point at infinity, the approximation is written in terms of an inverse power x^{−1} with the indicial equation
Example 6C:
The inicial equation ν² + 4ν + 13 = 0 for Euler's equation
\[
x^2 y'' +5x\,y' + 13\,y = 0
\]
has two complex conjugate roots ν = −2 ±3j, where j is the unit vector in the positive vertical direction on the complex plane ℂ, with j² = −1. Hence its general solution becomes
\[
y (x) = C_1 x^{-2} \cos \left( 3\ln x \right) + C_2 x^{-2} \sin \left( 3\ln x \right) .
\]
Upon transformation x = t^{−1}, we obtain
\[
t^2 \ddot{u} + t \left( 2 - 5 \right) \dot{u} + 13\, u = 0 .
\]
are the derivative operator and the identical operator, respectively. It is also sometimes convenient to eliminate fractions and consider a similar operator:
are the derivative operator and the identical operator, respectively. It is also sometimes convenient to eliminate fractions and consider a similar operator:
are the derivative operator and the identical operator, respectively. It is also sometimes convenient to eliminate fractions and consider a similar operator:
Otherwise, the factorization system of algebraic equations leads to complex roots that we do not consider. Note that factorization \eqref{EqEuler.8} or \eqref{EqEuler.10} is not unique. The value n in these formulas is one of the roots of the indicial equation.
Example 7:
Consider the singular differential operator
To find its inverse, we need to solve the differential equation
\[
L \left[ x,\texttt{D} \right] y = w \qquad \Longrightarrow \qquad
x^5 \texttt{D}\, y = \left( x^5 \texttt{D}\, y \right)_{x= x_0} + \int_{x_0}^x t^5 w(t)\,{\text d} t .
\]
where primes correspond to derivatives: y' = dy/dx. Its indicial equation r² −2r + 1 = 0 has one double root r = 1. Solving the factorization system of equations
\[
m + 2n = -1, \qquad n \left( m+n-1 \right) = 1 ,
\]
with some real constants p, q, and given function
g(x). We denote the left-hand side of the above equation by
\( L \left[ y(x) \right] = y'' (x) + p\, y' (x)/x + q\,y(x)/x^2 , \) the linear differential operator with variable coefficients.
Since it contains the reciprocal of x, the origin x = 0 is a
singular point for the Euler equation. This means that we can obtain its solutions either for negative values of the argument, x < 0, or for positive
values x > 0. The initial condition can be chosen at any point other than the origin. Therefore, we consider the Euler equations only for positive values x > 0.
Grigorieva, E., Methods of Solving Sequence and Series Problems, Birkhäuser; 1st ed. 2016.
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