60.7.202 problem 1827 (book 6.236)
Internal
problem
ID
[11752]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
6,
non-linear
second
order
Problem
number
:
1827
(book
6.236)
Date
solved
:
Sunday, March 30, 2025 at 09:13:34 PM
CAS
classification
:
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\begin{align*} {y^{\prime \prime }}^{2}-a y-b&=0 \end{align*}
✓ Maple. Time used: 0.321 (sec). Leaf size: 206
ode:=diff(diff(y(x),x),x)^2-a*y(x)-b = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -\frac {b}{a} \\
a \sqrt {3}\, \int _{}^{y}\frac {1}{\sqrt {a \left (4 \textit {\_a} \sqrt {a \textit {\_a} +b}\, a +4 \sqrt {a \textit {\_a} +b}\, b -c_1 \right )}}d \textit {\_a} -x -c_2 &= 0 \\
-a \sqrt {3}\, \int _{}^{y}\frac {1}{\sqrt {a \left (4 \textit {\_a} \sqrt {a \textit {\_a} +b}\, a +4 \sqrt {a \textit {\_a} +b}\, b -c_1 \right )}}d \textit {\_a} -x -c_2 &= 0 \\
-a \sqrt {3}\, \int _{}^{y}\frac {1}{\sqrt {-a \left (4 \textit {\_a} \sqrt {a \textit {\_a} +b}\, a +4 \sqrt {a \textit {\_a} +b}\, b -c_1 \right )}}d \textit {\_a} -x -c_2 &= 0 \\
a \sqrt {3}\, \int _{}^{y}\frac {1}{\sqrt {-a \left (4 \textit {\_a} \sqrt {a \textit {\_a} +b}\, a +4 \sqrt {a \textit {\_a} +b}\, b -c_1 \right )}}d \textit {\_a} -x -c_2 &= 0 \\
\end{align*}
✓ Mathematica. Time used: 1.23 (sec). Leaf size: 201
ode=-b - a*y[x] + D[y[x],{x,2}]^2 == 0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
\text {Solve}\left [\frac {(a y(x)+b)^2 \left (1-\frac {4 (a y(x)+b)^{3/2}}{3 a c_1}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},\frac {4 (b+a y(x))^{3/2}}{3 a c_1}\right ){}^2}{a^2 \left (-\frac {4 (a y(x)+b)^{3/2}}{3 a}+c_1\right )}&=(x+c_2){}^2,y(x)\right ] \\
\text {Solve}\left [\frac {(a y(x)+b)^2 \left (1+\frac {4 (a y(x)+b)^{3/2}}{3 a c_1}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},-\frac {4 (b+a y(x))^{3/2}}{3 a c_1}\right ){}^2}{a^2 \left (\frac {4 (a y(x)+b)^{3/2}}{3 a}+c_1\right )}&=(x+c_2){}^2,y(x)\right ] \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq(-a*y(x) - b + Derivative(y(x), (x, 2))**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : solve: Cannot solve -a*y(x) - b + Derivative(y(x), (x, 2))**2