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
:
Monday, January 27, 2025 at 11:18:55 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*}
✓ Solution by Maple
Time used: 0.040 (sec). Leaf size: 349
dsolve(diff(y(x),x)^2+(a+6*y(x))*diff(y(x),x)+y(x)*(3*a+b+9*y(x)) = 0,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= -\frac {\operatorname {RootOf}\left (-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 )+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 )+18 c_{1} a +6 c_{1} b -18 a x -6 b x \right ) \left (\operatorname {RootOf}\left (-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 )+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 )+18 c_{1} a +6 c_{1} b -18 a x -6 b x \right )+2 a \right )}{4 b} \\
y \left (x \right ) &= -\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 a x -6 b x \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 a x -6 b x \right )}+2 a \right )}{4 b} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.681 (sec). Leaf size: 175
DSolve[(D[y[x],x])^2+(a+6*y[x])*D[y[x],x]+y[x]*(3*a+b+9*y[x])==0,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*}