82.8.12 problem 36-13
Internal
problem
ID
[21883]
Book
:
The
Differential
Equations
Problem
Solver.
VOL.
II.
M.
Fogiel
director.
REA,
NY.
1978.
ISBN
78-63609
Section
:
Chapter
36.
Nonlinear
differential
equations.
Page
1203
Problem
number
:
36-13
Date
solved
:
Thursday, October 02, 2025 at 08:03:28 PM
CAS
classification
:
[[_1st_order, _with_linear_symmetries], _dAlembert]
\begin{align*} y&=2 x y^{\prime }-{y^{\prime }}^{2} \end{align*}
✓ Maple. Time used: 0.011 (sec). Leaf size: 611
ode:=y(x) = 2*x*diff(y(x),x)-diff(y(x),x)^2;
dsolve(ode,y(x), singsol=all);
\begin{align*} \text {Solution too large to show}\end{align*}
✓ Mathematica. Time used: 60.109 (sec). Leaf size: 954
ode=y[x]==2*x*D[y[x],x]-D[y[x],x]^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {1}{4} \left (x^2+\frac {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}}}+\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}{72} \left (18 x^2-\frac {9 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}}}+9 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}{72} \left (18 x^2+\frac {9 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}}}-9 \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 {x^4+\left (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 ){}^{2/3}+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}}-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}{72} \left (18 x^2+\frac {9 \left (1+i \sqrt {3}\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}}}+9 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}{72} \left (18 x^2+\frac {9 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}}}-9 \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 ) \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-2*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 - sqrt(x**2 - y(x)) + Derivative(y(x), x) cannot be solved by the factorable group method