60.7.200 problem 1824 (book 6.233)
Internal
problem
ID
[11750]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
6,
non-linear
second
order
Problem
number
:
1824
(book
6.233)
Date
solved
:
Thursday, March 13, 2025 at 09:23:13 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} \left ({y^{\prime }}^{2}+a \left (x y^{\prime }-y\right )\right ) y^{\prime \prime }-b&=0 \end{align*}
✓ Maple. Time used: 0.125 (sec). Leaf size: 289
ode:=(diff(y(x),x)^2+a*(-y(x)+x*diff(y(x),x)))*diff(diff(y(x),x),x)-b = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -\frac {a \,x^{2}}{4}+\operatorname {RootOf}\left (-x +\int _{}^{\textit {\_Z}}\frac {\sqrt {\left (\textit {\_f}^{2} a^{2}-4 b \textit {\_f} +2 c_{1} \right ) \left (a \textit {\_f} +\sqrt {4 b \textit {\_f} -2 c_{1}}\right )}}{\textit {\_f}^{2} a^{2}-4 b \textit {\_f} +2 c_{1}}d \textit {\_f} +c_{2} \right ) \\
y &= -\frac {a \,x^{2}}{4}+\operatorname {RootOf}\left (-x +\int _{}^{\textit {\_Z}}\frac {\sqrt {\left (\textit {\_f}^{2} a^{2}-4 b \textit {\_f} +2 c_{1} \right ) \left (a \textit {\_f} -\sqrt {4 b \textit {\_f} -2 c_{1}}\right )}}{\textit {\_f}^{2} a^{2}-4 b \textit {\_f} +2 c_{1}}d \textit {\_f} +c_{2} \right ) \\
y &= -\frac {a \,x^{2}}{4}+\operatorname {RootOf}\left (-x -\int _{}^{\textit {\_Z}}\frac {\sqrt {\left (\textit {\_f}^{2} a^{2}-4 b \textit {\_f} +2 c_{1} \right ) \left (a \textit {\_f} +\sqrt {4 b \textit {\_f} -2 c_{1}}\right )}}{\textit {\_f}^{2} a^{2}-4 b \textit {\_f} +2 c_{1}}d \textit {\_f} +c_{2} \right ) \\
y &= -\frac {a \,x^{2}}{4}+\operatorname {RootOf}\left (-x -\int _{}^{\textit {\_Z}}\frac {\sqrt {\left (\textit {\_f}^{2} a^{2}-4 b \textit {\_f} +2 c_{1} \right ) \left (a \textit {\_f} -\sqrt {4 b \textit {\_f} -2 c_{1}}\right )}}{\textit {\_f}^{2} a^{2}-4 b \textit {\_f} +2 c_{1}}d \textit {\_f} +c_{2} \right ) \\
\end{align*}
✓ Mathematica. Time used: 0.489 (sec). Leaf size: 281
ode=-b + (D[y[x],x]^2 + a*(-y[x] + x*D[y[x],x]))*D[y[x],{x,2}] == 0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
\text {Solve}\left [-\int \frac {a \left (\frac {a x^2}{4}+y(x)\right )+\sqrt {4 b \left (\frac {a x^2}{4}+y(x)\right )-2 c_1}}{\sqrt {\left (a^2 \left (\frac {a x^2}{4}+y(x)\right )^2-4 b \left (\frac {a x^2}{4}+y(x)\right )+2 c_1\right ) \left (a \left (\frac {a x^2}{4}+y(x)\right )+\sqrt {4 b \left (\frac {a x^2}{4}+y(x)\right )-2 c_1}\right )}}d\left (\frac {a x^2}{4}+y(x)\right )&=-x+c_2,y(x)\right ] \\
\text {Solve}\left [\int \frac {a \left (\frac {a x^2}{4}+y(x)\right )+\sqrt {4 b \left (\frac {a x^2}{4}+y(x)\right )-2 c_1}}{\sqrt {\left (a^2 \left (\frac {a x^2}{4}+y(x)\right )^2-4 b \left (\frac {a x^2}{4}+y(x)\right )+2 c_1\right ) \left (a \left (\frac {a x^2}{4}+y(x)\right )+\sqrt {4 b \left (\frac {a x^2}{4}+y(x)\right )-2 c_1}\right )}}d\left (\frac {a x^2}{4}+y(x)\right )&=-x+c_2,y(x)\right ] \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq(-b + (a*(x*Derivative(y(x), x) - y(x)) + Derivative(y(x), x)**2)*Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -(-a*x*Derivative(y(x), (x, 2)) + sqrt((a**2*x**2*Derivative(y(x), (x, 2)) + 4*a*y(x)*Derivative(y(x), (x, 2)) + 4*b)*Derivative(y(x), (x, 2))))/(2*Derivative(y(x), (x, 2))) + Derivative(y(x), x) cannot be solved by the factorable group method