18.3.13 problem Problem 16.14
Internal
problem
ID
[3513]
Book
:
Mathematical
methods
for
physics
and
engineering,
Riley,
Hobson,
Bence,
second
edition,
2002
Section
:
Chapter
16,
Series
solutions
of
ODEs.
Section
16.6
Exercises,
page
550
Problem
number
:
Problem
16.14
Date
solved
:
Sunday, March 30, 2025 at 01:45:27 AM
CAS
classification
:
[_Laguerre]
\begin{align*} z y^{\prime \prime }+\left (1-z \right ) y^{\prime }+\lambda y&=0 \end{align*}
Using series method with expansion around
\begin{align*} 0 \end{align*}
✓ Maple. Time used: 0.025 (sec). Leaf size: 154
Order:=6;
ode:=z*diff(diff(y(z),z),z)+(1-z)*diff(y(z),z)+lambda*y(z) = 0;
dsolve(ode,y(z),type='series',z=0);
\[
y = \left (\left (2 \lambda +1\right ) z +\left (\frac {1}{4} \lambda +\frac {1}{4}-\frac {3}{4} \lambda ^{2}\right ) z^{2}+\left (-\frac {2}{9} \lambda ^{2}+\frac {1}{27} \lambda +\frac {1}{18}+\frac {11}{108} \lambda ^{3}\right ) z^{3}+\left (\frac {7}{192} \lambda ^{3}-\frac {167}{3456} \lambda ^{2}+\frac {1}{192} \lambda +\frac {1}{96}-\frac {25}{3456} \lambda ^{4}\right ) z^{4}+\left (\frac {719}{86400} \lambda ^{3}-\frac {61}{21600} \lambda ^{4}+\frac {137}{432000} \lambda ^{5}+\frac {1}{600}-\frac {37}{4320} \lambda ^{2}+\frac {1}{1500} \lambda \right ) z^{5}+\operatorname {O}\left (z^{6}\right )\right ) c_2 +\left (1-\lambda z +\frac {1}{4} \left (-1+\lambda \right ) \lambda z^{2}-\frac {1}{36} \left (\lambda -2\right ) \left (-1+\lambda \right ) \lambda z^{3}+\frac {1}{576} \left (\lambda -3\right ) \left (\lambda -2\right ) \left (-1+\lambda \right ) \lambda z^{4}-\frac {1}{14400} \left (\lambda -4\right ) \left (\lambda -3\right ) \left (\lambda -2\right ) \left (-1+\lambda \right ) \lambda z^{5}+\operatorname {O}\left (z^{6}\right )\right ) \left (c_2 \ln \left (z \right )+c_1 \right )
\]
✓ Mathematica. Time used: 0.005 (sec). Leaf size: 415
ode=z*D[y[z],{z,2}]+(1-z)*D[y[z],z]+\[Lambda]*y[z]==0;
ic={};
AsymptoticDSolveValue[{ode,ic},y[z],{z,0,5}]
\[
y(z)\to c_1 \left (-\frac {(\lambda -4) (\lambda -3) (\lambda -2) (\lambda -1) \lambda z^5}{14400}+\frac {1}{576} (\lambda -3) (\lambda -2) (\lambda -1) \lambda z^4-\frac {1}{36} (\lambda -2) (\lambda -1) \lambda z^3+\frac {1}{4} (\lambda -1) \lambda z^2-\lambda z+1\right )+c_2 \left (\frac {(\lambda -4) (\lambda -3) (\lambda -2) (\lambda -1) z^5}{14400}+\frac {(\lambda -4) (\lambda -3) (\lambda -2) \lambda z^5}{14400}+\frac {(\lambda -4) (\lambda -3) (\lambda -1) \lambda z^5}{14400}+\frac {(\lambda -4) (\lambda -2) (\lambda -1) \lambda z^5}{14400}+\frac {137 (\lambda -4) (\lambda -3) (\lambda -2) (\lambda -1) \lambda z^5}{432000}+\frac {(\lambda -3) (\lambda -2) (\lambda -1) \lambda z^5}{14400}-\frac {1}{576} (\lambda -3) (\lambda -2) (\lambda -1) z^4-\frac {1}{576} (\lambda -3) (\lambda -2) \lambda z^4-\frac {1}{576} (\lambda -3) (\lambda -1) \lambda z^4-\frac {25 (\lambda -3) (\lambda -2) (\lambda -1) \lambda z^4}{3456}-\frac {1}{576} (\lambda -2) (\lambda -1) \lambda z^4+\frac {1}{36} (\lambda -2) (\lambda -1) z^3+\frac {1}{36} (\lambda -2) \lambda z^3+\frac {11}{108} (\lambda -2) (\lambda -1) \lambda z^3+\frac {1}{36} (\lambda -1) \lambda z^3-\frac {1}{4} (\lambda -1) z^2-\frac {3}{4} (\lambda -1) \lambda z^2-\frac {\lambda z^2}{4}+\left (-\frac {(\lambda -4) (\lambda -3) (\lambda -2) (\lambda -1) \lambda z^5}{14400}+\frac {1}{576} (\lambda -3) (\lambda -2) (\lambda -1) \lambda z^4-\frac {1}{36} (\lambda -2) (\lambda -1) \lambda z^3+\frac {1}{4} (\lambda -1) \lambda z^2-\lambda z+1\right ) \log (z)+2 \lambda z+z\right )
\]
✓ Sympy. Time used: 0.938 (sec). Leaf size: 73
from sympy import *
z = symbols("z")
lambda_ = symbols("lambda_")
y = Function("y")
ode = Eq(lambda_*y(z) + z*Derivative(y(z), (z, 2)) + (1 - z)*Derivative(y(z), z),0)
ics = {}
dsolve(ode,func=y(z),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
\[
y{\left (z \right )} = C_{1} \left (- \frac {\lambda _{} z^{5} \left (\lambda _{} - 4\right ) \left (\lambda _{} - 3\right ) \left (\lambda _{} - 2\right ) \left (\lambda _{} - 1\right )}{14400} + \frac {\lambda _{} z^{4} \left (\lambda _{} - 3\right ) \left (\lambda _{} - 2\right ) \left (\lambda _{} - 1\right )}{576} - \frac {\lambda _{} z^{3} \left (\lambda _{} - 2\right ) \left (\lambda _{} - 1\right )}{36} + \frac {\lambda _{} z^{2} \left (\lambda _{} - 1\right )}{4} - \lambda _{} z + 1\right ) + O\left (z^{6}\right )
\]