23.4.154 problem 154
Internal
problem
ID
[6456]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Part
II.
Chapter
4.
THE
NONLINEAR
EQUATION
OF
SECOND
ORDER,
page
380
Problem
number
:
154
Date
solved
:
Tuesday, September 30, 2025 at 02:57:27 PM
CAS
classification
:
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\begin{align*} y y^{\prime \prime }&=b +a {y^{\prime }}^{2} \end{align*}
✓ Maple. Time used: 0.022 (sec). Leaf size: 64
ode:=y(x)*diff(diff(y(x),x),x) = b+a*diff(y(x),x)^2;
dsolve(ode,y(x), singsol=all);
\begin{align*}
a \int _{}^{y}\frac {1}{\sqrt {a \left (c_1 \,\textit {\_a}^{2 a}-b \right )}}d \textit {\_a} -x -c_2 &= 0 \\
-a \int _{}^{y}\frac {1}{\sqrt {a \left (c_1 \,\textit {\_a}^{2 a}-b \right )}}d \textit {\_a} -x -c_2 &= 0 \\
\end{align*}
✓ Mathematica. Time used: 0.982 (sec). Leaf size: 628
ode=y[x]*D[y[x],{x,2}] == b + a*D[y[x],x]^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \text {InverseFunction}\left [-\frac {\text {$\#$1} \sqrt {a} \sqrt {\frac {b-\text {$\#$1}^{2 a} e^{2 a c_1}}{b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2 a},1+\frac {1}{2 a},\frac {e^{2 a c_1} \text {$\#$1}^{2 a}}{b}\right )}{\sqrt {-b+\text {$\#$1}^{2 a} e^{2 a c_1}}}\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\frac {\text {$\#$1} \sqrt {a} \sqrt {\frac {b-\text {$\#$1}^{2 a} e^{2 a c_1}}{b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2 a},1+\frac {1}{2 a},\frac {e^{2 a c_1} \text {$\#$1}^{2 a}}{b}\right )}{\sqrt {-b+\text {$\#$1}^{2 a} e^{2 a c_1}}}\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [-\frac {\text {$\#$1} \sqrt {a} \sqrt {\frac {b-\text {$\#$1}^{2 a} e^{2 a (-c_1)}}{b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2 a},1+\frac {1}{2 a},\frac {e^{2 a (-c_1)} \text {$\#$1}^{2 a}}{b}\right )}{\sqrt {-b+\text {$\#$1}^{2 a} e^{2 a (-c_1)}}}\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\frac {\text {$\#$1} \sqrt {a} \sqrt {\frac {b-\text {$\#$1}^{2 a} e^{2 a (-c_1)}}{b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2 a},1+\frac {1}{2 a},\frac {e^{2 a (-c_1)} \text {$\#$1}^{2 a}}{b}\right )}{\sqrt {-b+\text {$\#$1}^{2 a} e^{2 a (-c_1)}}}\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [-\frac {\text {$\#$1} \sqrt {a} \sqrt {\frac {b-\text {$\#$1}^{2 a} e^{2 a c_1}}{b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2 a},1+\frac {1}{2 a},\frac {e^{2 a c_1} \text {$\#$1}^{2 a}}{b}\right )}{\sqrt {-b+\text {$\#$1}^{2 a} e^{2 a c_1}}}\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\frac {\text {$\#$1} \sqrt {a} \sqrt {\frac {b-\text {$\#$1}^{2 a} e^{2 a c_1}}{b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2 a},1+\frac {1}{2 a},\frac {e^{2 a c_1} \text {$\#$1}^{2 a}}{b}\right )}{\sqrt {-b+\text {$\#$1}^{2 a} e^{2 a c_1}}}\&\right ][x+c_2] \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq(-a*Derivative(y(x), x)**2 - b + y(x)*Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -sqrt((-b + y(x)*Derivative(y(x), (x, 2)))/a) + Derivative(y(x),