56.3.2 problem 2

Internal problem ID [8860]
Book : Own collection of miscellaneous problems
Section : section 3.0
Problem number : 2
Date solved : Monday, January 27, 2025 at 05:07:18 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*}

✓ Solution by Maple

Time used: 1.025 (sec). Leaf size: 65

dsolve([diff(w(z),z) = -1/2 - sqrt(1/4 - 3*w(z)),w(1) = -1],w(z), singsol=all)
 

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

✓ Solution by Mathematica

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

DSolve[{D[w[z],z] == -1/2 - Sqrt[1/4 - 3*w[z]],{w[1] == -1}},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*}