12.3.8 problem Problem 16.10

Internal problem ID [3508]
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 16.10
Date solved : Tuesday, September 30, 2025 at 06:40:58 AM
CAS classification : [_Jacobi]

\begin{align*} z \left (1-z \right ) 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: 0.026 (sec). Leaf size: 154

Order:=6; 
ode:=z*(1-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);
 

\[ y = \left (2 \lambda z +\left (\frac {1}{4} \lambda -\frac {3}{4} \lambda ^{2}\right ) z^{2}+\left (-\frac {37}{108} \lambda ^{2}+\frac {2}{27} \lambda +\frac {11}{108} \lambda ^{3}\right ) z^{3}+\left (\frac {139}{1728} \lambda ^{3}-\frac {649}{3456} \lambda ^{2}+\frac {1}{32} \lambda -\frac {25}{3456} \lambda ^{4}\right ) z^{4}+\left (-\frac {13}{1600} \lambda ^{4}+\frac {8467}{144000} \lambda ^{3}-\frac {2527}{21600} \lambda ^{2}+\frac {2}{125} \lambda +\frac {137}{432000} \lambda ^{5}\right ) z^{5}+\operatorname {O}\left (z^{6}\right )\right ) c_2 +\left (1-\lambda z +\frac {1}{4} \lambda \left (\lambda -1\right ) z^{2}-\frac {1}{36} \lambda \left (\lambda ^{2}-5 \lambda +4\right ) z^{3}+\frac {1}{576} \lambda \left (\lambda ^{3}-14 \lambda ^{2}+49 \lambda -36\right ) z^{4}-\frac {1}{14400} \lambda \left (\lambda -1\right ) \left (\lambda -4\right ) \left (\lambda -16\right ) \left (\lambda -9\right ) z^{5}+\operatorname {O}\left (z^{6}\right )\right ) \left (c_2 \ln \left (z \right )+c_1 \right ) \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 940

ode=z*(1-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}]
 

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Sympy. Time used: 0.531 (sec). Leaf size: 82

from sympy import * 
z = symbols("z") 
lambda_ = symbols("lambda_") 
y = Function("y") 
ode = Eq(lambda_*y(z) + z*(1 - 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)
 

\[ y{\left (z \right )} = C_{1} \left (\frac {\lambda _{} z^{5} \left (\lambda _{}^{2} \left (- \lambda _{}^{2} - 5\right ) - 25 \lambda _{}^{2} - 89\right )}{14400} + \frac {\lambda _{} z^{3} \left (- \lambda _{}^{2} - 5\right )}{36} - \lambda _{} z + \frac {z^{4} \left (- \lambda _{}^{2} \left (- \lambda _{}^{2} - 5\right ) + 9 \lambda _{}^{2} + 9\right )}{576} + \frac {z^{2} \left (\lambda _{}^{2} + 1\right )}{4} + 1\right ) + O\left (z^{6}\right ) \]