47.2.3 problem 10
Internal
problem
ID
[9742]
Book
:
Elementary
differential
equations.
By
Earl
D.
Rainville,
Phillip
E.
Bedient.
Macmilliam
Publishing
Co.
NY.
6th
edition.
1981.
Section
:
CHAPTER
16.
Nonlinear
equations.
Section
97.
The
p-discriminant
equation.
EXERCISES
Page
314
Problem
number
:
10
Date
solved
:
Tuesday, September 30, 2025 at 06:32:36 PM
CAS
classification
:
[[_1st_order, _with_linear_symmetries], _dAlembert]
\begin{align*} {y^{\prime }}^{2}-x y^{\prime }-y&=0 \end{align*}
✓ Maple. Time used: 0.035 (sec). Leaf size: 77
ode:=diff(y(x),x)^2-x*diff(y(x),x)-y(x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
\frac {c_1}{\sqrt {2 x -2 \sqrt {x^{2}+4 y}}}+\frac {2 x}{3}+\frac {\sqrt {x^{2}+4 y}}{3} &= 0 \\
\frac {c_1}{\sqrt {2 x +2 \sqrt {x^{2}+4 y}}}+\frac {2 x}{3}-\frac {\sqrt {x^{2}+4 y}}{3} &= 0 \\
\end{align*}
✓ Mathematica. Time used: 60.069 (sec). Leaf size: 1003
ode=(D[y[x],x])^2-x*D[y[x],x]-y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {\left (x^2+\sqrt [3]{-x^6+20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}}\right ){}^2+8 e^{3 c_1} x}{4 \sqrt [3]{-x^6+20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}}}\\ y(x)&\to \frac {1}{8} \left (4 x^2-\frac {i \left (\sqrt {3}-i\right ) x \left (x^3+8 e^{3 c_1}\right )}{\sqrt [3]{-x^6+20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}}}+i \left (\sqrt {3}+i\right ) \sqrt [3]{-x^6+20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}}\right )\\ y(x)&\to \frac {1}{8} \left (4 x^2+\frac {i \left (\sqrt {3}+i\right ) x \left (x^3+8 e^{3 c_1}\right )}{\sqrt [3]{-x^6+20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}}}-\left (1+i \sqrt {3}\right ) \sqrt [3]{-x^6+20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}}\right )\\ y(x)&\to \frac {2 \sqrt [3]{2} x^4+2^{2/3} \left (-2 x^6-10 e^{3 c_1} x^3+\sqrt {e^{3 c_1} \left (4 x^3+e^{3 c_1}\right ){}^3}+e^{6 c_1}\right ){}^{2/3}+4 x^2 \sqrt [3]{-2 x^6-10 e^{3 c_1} x^3+\sqrt {e^{3 c_1} \left (4 x^3+e^{3 c_1}\right ){}^3}+e^{6 c_1}}-4 \sqrt [3]{2} e^{3 c_1} x}{8 \sqrt [3]{-2 x^6-10 e^{3 c_1} x^3+\sqrt {e^{3 c_1} \left (4 x^3+e^{3 c_1}\right ){}^3}+e^{6 c_1}}}\\ y(x)&\to \frac {1}{16} \left (8 x^2+\frac {2 \sqrt [3]{2} \left (1+i \sqrt {3}\right ) x \left (-x^3+2 e^{3 c_1}\right )}{\sqrt [3]{-2 x^6-10 e^{3 c_1} x^3+\sqrt {e^{3 c_1} \left (4 x^3+e^{3 c_1}\right ){}^3}+e^{6 c_1}}}+i 2^{2/3} \left (\sqrt {3}+i\right ) \sqrt [3]{-2 x^6-10 e^{3 c_1} x^3+\sqrt {e^{3 c_1} \left (4 x^3+e^{3 c_1}\right ){}^3}+e^{6 c_1}}\right )\\ y(x)&\to \frac {1}{16} \left (8 x^2+\frac {2 i \sqrt [3]{2} \left (\sqrt {3}+i\right ) x \left (x^3-2 e^{3 c_1}\right )}{\sqrt [3]{-2 x^6-10 e^{3 c_1} x^3+\sqrt {e^{3 c_1} \left (4 x^3+e^{3 c_1}\right ){}^3}+e^{6 c_1}}}-2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{-2 x^6-10 e^{3 c_1} x^3+\sqrt {e^{3 c_1} \left (4 x^3+e^{3 c_1}\right ){}^3}+e^{6 c_1}}\right ) \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x*Derivative(y(x), x) - y(x) + Derivative(y(x), x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -x/2 - sqrt(x**2 + 4*y(x))/2 + Derivative(y(x), x) cannot be solved by the factorable group method