29.32.19 problem 953
Internal
problem
ID
[5531]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
32
Problem
number
:
953
Date
solved
:
Monday, January 27, 2025 at 11:35:41 AM
CAS
classification
:
[_quadrature]
\begin{align*} y {y^{\prime }}^{2}+y&=a \end{align*}
✓ Solution by Maple
Time used: 0.053 (sec). Leaf size: 339
dsolve(y(x)*diff(y(x),x)^2+y(x) = a,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= a \\
y \left (x \right ) &= \frac {\left (\operatorname {RootOf}\left (\left (\cos \left (\textit {\_Z} \right ) a +\textit {\_Z} a +2 c_{1} -2 x \right ) \left (-\cos \left (\textit {\_Z} \right ) a +\textit {\_Z} a +2 c_{1} -2 x \right )\right ) a -2 x +2 c_{1} \right ) \tan \left (\operatorname {RootOf}\left (\left (\cos \left (\textit {\_Z} \right ) a +\textit {\_Z} a +2 c_{1} -2 x \right ) \left (-\cos \left (\textit {\_Z} \right ) a +\textit {\_Z} a +2 c_{1} -2 x \right )\right )\right )}{2}+\frac {a}{2} \\
y \left (x \right ) &= \frac {\left (-\operatorname {RootOf}\left (\left (\cos \left (\textit {\_Z} \right ) a +\textit {\_Z} a +2 c_{1} -2 x \right ) \left (-\cos \left (\textit {\_Z} \right ) a +\textit {\_Z} a +2 c_{1} -2 x \right )\right ) a +2 x -2 c_{1} \right ) \tan \left (\operatorname {RootOf}\left (\left (\cos \left (\textit {\_Z} \right ) a +\textit {\_Z} a +2 c_{1} -2 x \right ) \left (-\cos \left (\textit {\_Z} \right ) a +\textit {\_Z} a +2 c_{1} -2 x \right )\right )\right )}{2}+\frac {a}{2} \\
y \left (x \right ) &= \frac {\left (\operatorname {RootOf}\left (\left (\cos \left (\textit {\_Z} \right ) a -\textit {\_Z} a +2 c_{1} -2 x \right ) \left (-\cos \left (\textit {\_Z} \right ) a -\textit {\_Z} a +2 c_{1} -2 x \right )\right ) a +2 x -2 c_{1} \right ) \tan \left (\operatorname {RootOf}\left (\left (\cos \left (\textit {\_Z} \right ) a -\textit {\_Z} a +2 c_{1} -2 x \right ) \left (-\cos \left (\textit {\_Z} \right ) a -\textit {\_Z} a +2 c_{1} -2 x \right )\right )\right )}{2}+\frac {a}{2} \\
y \left (x \right ) &= \frac {\left (-\operatorname {RootOf}\left (\left (\cos \left (\textit {\_Z} \right ) a -\textit {\_Z} a +2 c_{1} -2 x \right ) \left (-\cos \left (\textit {\_Z} \right ) a -\textit {\_Z} a +2 c_{1} -2 x \right )\right ) a -2 x +2 c_{1} \right ) \tan \left (\operatorname {RootOf}\left (\left (\cos \left (\textit {\_Z} \right ) a -\textit {\_Z} a +2 c_{1} -2 x \right ) \left (-\cos \left (\textit {\_Z} \right ) a -\textit {\_Z} a +2 c_{1} -2 x \right )\right )\right )}{2}+\frac {a}{2} \\
\end{align*}
✓ Solution by Mathematica
Time used: 1.027 (sec). Leaf size: 142
DSolve[y[x] (D[y[x],x])^2+y[x]==a,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \text {InverseFunction}\left [\frac {a^{3/2} \sqrt {1-\frac {\text {$\#$1}}{a}} \arcsin \left (\frac {\sqrt {\text {$\#$1}}}{\sqrt {a}}\right )+\sqrt {\text {$\#$1}} (\text {$\#$1}-a)}{\sqrt {a-\text {$\#$1}}}\&\right ][-x+c_1] \\
y(x)\to \text {InverseFunction}\left [\frac {a^{3/2} \sqrt {1-\frac {\text {$\#$1}}{a}} \arcsin \left (\frac {\sqrt {\text {$\#$1}}}{\sqrt {a}}\right )+\sqrt {\text {$\#$1}} (\text {$\#$1}-a)}{\sqrt {a-\text {$\#$1}}}\&\right ][x+c_1] \\
y(x)\to a \\
\end{align*}