79.15.1 problem (a)

Internal problem ID [21117]
Book : Ordinary Differential Equations. By Wolfgang Walter. Graduate texts in Mathematics. Springer. NY. QA372.W224 1998
Section : Chapter V. Complex Linear Systems. Excercise VIII at page 221
Problem number : (a)
Date solved : Sunday, October 12, 2025 at 05:51:27 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d z}w_{1} \left (z \right )&=w_{2} \left (z \right )\\ \frac {d}{d z}w_{2} \left (z \right )&=\frac {a w_{1} \left (z \right )}{z^{2}} \end{align*}

✓ Maple. Time used: 0.031 (sec). Leaf size: 94

ode:=[diff(w__1(z),z) = w__2(z), diff(w__2(z),z) = a*w__1(z)/z^2]; 
dsolve(ode);
 

\begin{align*} w_{1} \left (z \right ) &= c_1 \,z^{\frac {1}{2}+\frac {\sqrt {1+4 a}}{2}}+c_2 \,z^{\frac {1}{2}-\frac {\sqrt {1+4 a}}{2}} \\ w_{2} \left (z \right ) &= \frac {c_1 \,z^{\frac {1}{2}+\frac {\sqrt {1+4 a}}{2}} \left (1+\sqrt {1+4 a}\right )+c_2 \,z^{\frac {1}{2}-\frac {\sqrt {1+4 a}}{2}} \left (1-\sqrt {1+4 a}\right )}{2 z} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 128

ode={D[w1[z],z]==w2[z],D[w2[z],z]==a*w1[z]/z^2}; 
ic={}; 
DSolve[{ode,ic},{w1[z],w2[z]},z,IncludeSingularSolutions->True]
 

\begin{align*} \text {w1}(z)&\to z^{\frac {1}{2}-\frac {1}{2} i \sqrt {-4 a-1}} \left (c_2 z^{i \sqrt {-4 a-1}}+c_1\right )\\ \text {w2}(z)&\to \frac {1}{2} z^{-\frac {1}{2}-\frac {1}{2} i \sqrt {-4 a-1}} \left (\left (1+i \sqrt {-4 a-1}\right ) c_2 z^{i \sqrt {-4 a-1}}+\left (1-i \sqrt {-4 a-1}\right ) c_1\right ) \end{align*}

✗ Sympy

from sympy import * 
z = symbols("z") 
a = symbols("a") 
w1 = Function("w1") 
w2 = Function("w2") 
ode=[Eq(-w2(z) + Derivative(w1(z), z),0),Eq(-a*w1(z)/z**2 + Derivative(w2(z), z),0)] 
ics = {} 
dsolve(ode,func=[w1(z),w2(z)],ics=ics)
 

ValueError : The function cannot be automatically detected for nan.