29.28.18 problem 816
Internal
problem
ID
[5399]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
28
Problem
number
:
816
Date
solved
:
Sunday, March 30, 2025 at 08:07:26 AM
CAS
classification
:
[_quadrature]
\begin{align*} {y^{\prime }}^{2}+\left (a +6 y\right ) y^{\prime }+y \left (3 a +b +9 y\right )&=0 \end{align*}
✓ Maple. Time used: 1.630 (sec). Leaf size: 349
ode:=diff(y(x),x)^2+(a+6*y(x))*diff(y(x),x)+y(x)*(3*a+b+9*y(x)) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -\frac {{\mathrm e}^{\operatorname {RootOf}\left (-3 a \ln \left (-\frac {1}{b}\right )-3 a \ln \left (-\frac {\left (-3 \,{\mathrm e}^{\textit {\_Z}}+6 a +2 b \right )^{2}}{b}\right )-2 b \ln \left (-\frac {\left (-3 \,{\mathrm e}^{\textit {\_Z}}+6 a +2 b \right )^{2}}{b}\right )+12 a \ln \left (2\right )+4 b \ln \left (2\right )+18 c_1 a +6 c_1 b -6 \textit {\_Z} a -18 x a -6 x b \right )} \left ({\mathrm e}^{\operatorname {RootOf}\left (-3 a \ln \left (-\frac {1}{b}\right )-3 a \ln \left (-\frac {\left (-3 \,{\mathrm e}^{\textit {\_Z}}+6 a +2 b \right )^{2}}{b}\right )-2 b \ln \left (-\frac {\left (-3 \,{\mathrm e}^{\textit {\_Z}}+6 a +2 b \right )^{2}}{b}\right )+12 a \ln \left (2\right )+4 b \ln \left (2\right )+18 c_1 a +6 c_1 b -6 \textit {\_Z} a -18 x a -6 x b \right )}-2 a \right )}{4 b} \\
y &= -\frac {\operatorname {RootOf}\left (3 a \ln \left (-\frac {b}{\left (3 \textit {\_Z} +2 b \right )^{2}}\right )+2 b \ln \left (-\frac {b}{\left (3 \textit {\_Z} +2 b \right )^{2}}\right )-3 a \ln \left (-\frac {\left (\textit {\_Z} -2 a \right )^{2}}{b}\right )+12 a \ln \left (2\right )+4 b \ln \left (2\right )+18 c_1 a +6 c_1 b -18 x a -6 x b \right ) \left (\operatorname {RootOf}\left (3 a \ln \left (-\frac {b}{\left (3 \textit {\_Z} +2 b \right )^{2}}\right )+2 b \ln \left (-\frac {b}{\left (3 \textit {\_Z} +2 b \right )^{2}}\right )-3 a \ln \left (-\frac {\left (\textit {\_Z} -2 a \right )^{2}}{b}\right )+12 a \ln \left (2\right )+4 b \ln \left (2\right )+18 c_1 a +6 c_1 b -18 x a -6 x b \right )-2 a \right )}{4 b} \\
\end{align*}
✓ Mathematica. Time used: 0.681 (sec). Leaf size: 175
ode=(D[y[x],x])^2+(a+6*y[x])*D[y[x],x]+y[x]*(3*a+b+9*y[x])==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \text {InverseFunction}\left [\frac {3 a \log \left (\sqrt {a^2-4 \text {$\#$1} b}+a\right )+(3 a+2 b) \log \left (3 \sqrt {a^2-4 \text {$\#$1} b}-3 a-2 b\right )}{6 (3 a+b)}\&\right ]\left [-\frac {x}{2}+c_1\right ] \\
y(x)\to \text {InverseFunction}\left [-\frac {3 a \log \left (\sqrt {a^2-4 \text {$\#$1} b}-a\right )+(3 a+2 b) \log \left (3 \sqrt {a^2-4 \text {$\#$1} b}+3 a+2 b\right )}{6 (3 a+b)}\&\right ]\left [\frac {x}{2}+c_1\right ] \\
y(x)\to 0 \\
y(x)\to \frac {1}{9} (-3 a-b) \\
\end{align*}
✓ Sympy. Time used: 12.048 (sec). Leaf size: 184
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq((a + 6*y(x))*Derivative(y(x), x) + (3*a + b + 9*y(x))*y(x) + Derivative(y(x), x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ \begin {cases} - \frac {- \frac {a b \log {\left (a + \sqrt {a^{2} - 4 b y{\left (x \right )}} \right )}}{3 a + b} - \frac {b \left (3 a + 2 b\right ) \log {\left (3 a + 2 b - 3 \sqrt {a^{2} - 4 b y{\left (x \right )}} \right )}}{3 \left (3 a + b\right )}}{2 b} & \text {for}\: b \neq 0 \\\frac {\log {\left (a + \sqrt {a^{2}} + 6 y{\left (x \right )} \right )}}{6} & \text {otherwise} \end {cases} = C_{1} - \frac {x}{2}, \ \begin {cases} - \frac {- \frac {a b \log {\left (a - \sqrt {a^{2} - 4 b y{\left (x \right )}} \right )}}{3 a + b} - \frac {b \left (3 a + 2 b\right ) \log {\left (3 a + 2 b + 3 \sqrt {a^{2} - 4 b y{\left (x \right )}} \right )}}{3 \left (3 a + b\right )}}{2 b} & \text {for}\: b \neq 0 \\\frac {\log {\left (a - \sqrt {a^{2}} + 6 y{\left (x \right )} \right )}}{6} & \text {otherwise} \end {cases} = C_{1} - \frac {x}{2}\right ]
\]