56.3.2 problem 2

Internal problem ID [8860]
Book : Own collection of miscellaneous problems
Section : section 3.0
Problem number : 2
Date solved : Sunday, March 30, 2025 at 01:44:49 PM
CAS classification : [_quadrature]

\begin{align*} w^{\prime }&=-\frac {1}{2}-\frac {\sqrt {1-12 w}}{2} \end{align*}

With initial conditions

\begin{align*} w \left (1\right )&=-1 \end{align*}

✓ Maple. Time used: 0.626 (sec). Leaf size: 65

ode:=diff(w(z),z) = -1/2-1/2*(1-12*w(z))^(1/2); 
ic:=w(1) = -1; 
dsolve([ode,ic],w(z), singsol=all);
 

\[ w = \operatorname {RootOf}\left (-i \pi -2 \sqrt {1-12 \textit {\_Z}}+\ln \left (\textit {\_Z} \right )-\ln \left (-1+\sqrt {1-12 \textit {\_Z}}\right )+\ln \left (1+\sqrt {1-12 \textit {\_Z}}\right )-\ln \left (1+\sqrt {13}\right )+\ln \left (-1+\sqrt {13}\right )+2 \sqrt {13}+6 z -6\right ) \]

✓ Mathematica. Time used: 11.723 (sec). Leaf size: 105

ode=D[w[z],z] == -1/2 - Sqrt[1/4 - 3*w[z]]; 
ic={w[1] == -1}; 
DSolve[{ode,ic},w[z],z,IncludeSingularSolutions->True]
 

\begin{align*} w(z)\to -\frac {1}{12} W\left (\left (\sqrt {13}-1\right ) e^{-3 z+\sqrt {13}+2}\right ) \left (W\left (\left (\sqrt {13}-1\right ) e^{-3 z+\sqrt {13}+2}\right )+2\right ) \\ w(z)\to -\frac {1}{12} W\left (\left (\sqrt {13}-1\right ) e^{-3 z+\sqrt {13}+2}\right ) \left (W\left (\left (\sqrt {13}-1\right ) e^{-3 z+\sqrt {13}+2}\right )+2\right ) \\ \end{align*}

✗ Sympy

from sympy import * 
z = symbols("z") 
w = Function("w") 
ode = Eq(sqrt(1 - 12*w(z))/2 + Derivative(w(z), z) + 1/2,0) 
ics = {w(1): -1} 
dsolve(ode,func=w(z),ics=ics)
 

NotImplementedError : Initial conditions produced too many solutions for constants