23.2.201 problem 206
Internal
problem
ID
[5556]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Part
II.
Chapter
2.
THE
DIFFERENTIAL
EQUATION
IS
OF
FIRST
ORDER
AND
OF
SECOND
OR
HIGHER
DEGREE,
page
278
Problem
number
:
206
Date
solved
:
Tuesday, September 30, 2025 at 12:53:12 PM
CAS
classification
:
[_quadrature]
\begin{align*} y {y^{\prime }}^{2}+y&=a \end{align*}
✓ Maple. Time used: 0.132 (sec). Leaf size: 339
ode:=y(x)*diff(y(x),x)^2+y(x) = a;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= a \\
y &= \frac {\left (\operatorname {RootOf}\left (\left (a \cos \left (\textit {\_Z} \right )+\textit {\_Z} a +2 c_1 -2 x \right ) \left (-a \cos \left (\textit {\_Z} \right )+\textit {\_Z} a +2 c_1 -2 x \right )\right ) a -2 x +2 c_1 \right ) \tan \left (\operatorname {RootOf}\left (\left (a \cos \left (\textit {\_Z} \right )+\textit {\_Z} a +2 c_1 -2 x \right ) \left (-a \cos \left (\textit {\_Z} \right )+\textit {\_Z} a +2 c_1 -2 x \right )\right )\right )}{2}+\frac {a}{2} \\
y &= \frac {\left (-\operatorname {RootOf}\left (\left (a \cos \left (\textit {\_Z} \right )+\textit {\_Z} a +2 c_1 -2 x \right ) \left (-a \cos \left (\textit {\_Z} \right )+\textit {\_Z} a +2 c_1 -2 x \right )\right ) a +2 x -2 c_1 \right ) \tan \left (\operatorname {RootOf}\left (\left (a \cos \left (\textit {\_Z} \right )+\textit {\_Z} a +2 c_1 -2 x \right ) \left (-a \cos \left (\textit {\_Z} \right )+\textit {\_Z} a +2 c_1 -2 x \right )\right )\right )}{2}+\frac {a}{2} \\
y &= \frac {\left (\operatorname {RootOf}\left (\left (a \cos \left (\textit {\_Z} \right )-\textit {\_Z} a +2 c_1 -2 x \right ) \left (-a \cos \left (\textit {\_Z} \right )-\textit {\_Z} a +2 c_1 -2 x \right )\right ) a +2 x -2 c_1 \right ) \tan \left (\operatorname {RootOf}\left (\left (a \cos \left (\textit {\_Z} \right )-\textit {\_Z} a +2 c_1 -2 x \right ) \left (-a \cos \left (\textit {\_Z} \right )-\textit {\_Z} a +2 c_1 -2 x \right )\right )\right )}{2}+\frac {a}{2} \\
y &= \frac {\left (-\operatorname {RootOf}\left (\left (a \cos \left (\textit {\_Z} \right )-\textit {\_Z} a +2 c_1 -2 x \right ) \left (-a \cos \left (\textit {\_Z} \right )-\textit {\_Z} a +2 c_1 -2 x \right )\right ) a -2 x +2 c_1 \right ) \tan \left (\operatorname {RootOf}\left (\left (a \cos \left (\textit {\_Z} \right )-\textit {\_Z} a +2 c_1 -2 x \right ) \left (-a \cos \left (\textit {\_Z} \right )-\textit {\_Z} a +2 c_1 -2 x \right )\right )\right )}{2}+\frac {a}{2} \\
\end{align*}
✓ Mathematica. Time used: 0.606 (sec). Leaf size: 142
ode=y[x] (D[y[x],x])^2+y[x]==a;
ic={};
DSolve[{ode,ic},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*}
✓ Sympy. Time used: 1.146 (sec). Leaf size: 218
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(-a + y(x)*Derivative(y(x), x)**2 + y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ \begin {cases} - i \sqrt {a} \sqrt {-1 + \frac {y{\left (x \right )}}{a}} \sqrt {y{\left (x \right )}} - i a \operatorname {acosh}{\left (\frac {\sqrt {y{\left (x \right )}}}{\sqrt {a}} \right )} & \text {for}\: \left |{\frac {y{\left (x \right )}}{a}}\right | > 1 \\- \frac {\sqrt {a} \sqrt {y{\left (x \right )}}}{\sqrt {1 - \frac {y{\left (x \right )}}{a}}} + a \operatorname {asin}{\left (\frac {\sqrt {y{\left (x \right )}}}{\sqrt {a}} \right )} + \frac {y^{\frac {3}{2}}{\left (x \right )}}{\sqrt {a} \sqrt {1 - \frac {y{\left (x \right )}}{a}}} & \text {otherwise} \end {cases} = C_{1} - x, \ \begin {cases} - i \sqrt {a} \sqrt {-1 + \frac {y{\left (x \right )}}{a}} \sqrt {y{\left (x \right )}} - i a \operatorname {acosh}{\left (\frac {\sqrt {y{\left (x \right )}}}{\sqrt {a}} \right )} & \text {for}\: \left |{\frac {y{\left (x \right )}}{a}}\right | > 1 \\- \frac {\sqrt {a} \sqrt {y{\left (x \right )}}}{\sqrt {1 - \frac {y{\left (x \right )}}{a}}} + a \operatorname {asin}{\left (\frac {\sqrt {y{\left (x \right )}}}{\sqrt {a}} \right )} + \frac {y^{\frac {3}{2}}{\left (x \right )}}{\sqrt {a} \sqrt {1 - \frac {y{\left (x \right )}}{a}}} & \text {otherwise} \end {cases} = C_{1} + x\right ]
\]