71.1.64 problem 83 (page 120)
Internal
problem
ID
[19240]
Book
:
V.V.
Stepanov,
A
course
of
differential
equations
(in
Russian),
GIFML.
Moscow
(1958)
Section
:
All
content
Problem
number
:
83
(page
120)
Date
solved
:
Thursday, October 02, 2025 at 03:51:24 PM
CAS
classification
:
[[_1st_order, _with_linear_symmetries], _dAlembert]
\begin{align*} {y^{\prime }}^{2}+2 x y^{\prime }+2 y&=0 \end{align*}
✓ Maple. Time used: 0.028 (sec). Leaf size: 77
ode:=diff(y(x),x)^2+2*x*diff(y(x),x)+2*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
-\frac {c_1}{\sqrt {-x -\sqrt {x^{2}-2 y}}}+\frac {2 x}{3}-\frac {\sqrt {x^{2}-2 y}}{3} &= 0 \\
-\frac {c_1}{\sqrt {-x +\sqrt {x^{2}-2 y}}}+\frac {2 x}{3}+\frac {\sqrt {x^{2}-2 y}}{3} &= 0 \\
\end{align*}
✓ Mathematica. Time used: 60.116 (sec). Leaf size: 1000
ode=D[y[x],x]^2+2*x*D[y[x],x]+2*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+10 e^{3 c_1} x^3+2 \sqrt {e^{3 c_1} \left (2 x^3+e^{3 c_1}\right ){}^3}-2 e^{6 c_1}}\right ){}^2-4 e^{3 c_1} x}{2 \sqrt [3]{x^6+10 e^{3 c_1} x^3+2 \sqrt {e^{3 c_1} \left (2 x^3+e^{3 c_1}\right ){}^3}-2 e^{6 c_1}}}\\ y(x)&\to \frac {1}{4} \left (-4 x^2+\frac {\left (1+i \sqrt {3}\right ) x \left (-x^3+4 e^{3 c_1}\right )}{\sqrt [3]{x^6+10 e^{3 c_1} x^3+2 \sqrt {e^{3 c_1} \left (2 x^3+e^{3 c_1}\right ){}^3}-2 e^{6 c_1}}}+i \left (\sqrt {3}+i\right ) \sqrt [3]{x^6+10 e^{3 c_1} x^3+2 \sqrt {e^{3 c_1} \left (2 x^3+e^{3 c_1}\right ){}^3}-2 e^{6 c_1}}\right )\\ y(x)&\to -x^2+\frac {i \left (\sqrt {3}+i\right ) x \left (x^3-4 e^{3 c_1}\right )}{4 \sqrt [3]{x^6+10 e^{3 c_1} x^3+2 \sqrt {e^{3 c_1} \left (2 x^3+e^{3 c_1}\right ){}^3}-2 e^{6 c_1}}}-\frac {1}{4} \left (1+i \sqrt {3}\right ) \sqrt [3]{x^6+10 e^{3 c_1} x^3+2 \sqrt {e^{3 c_1} \left (2 x^3+e^{3 c_1}\right ){}^3}-2 e^{6 c_1}}\\ y(x)&\to \frac {2 \sqrt [3]{2} x^4+\left (4 x^6+20 e^{3 c_1} x^3+2 \sqrt {e^{3 c_1} \left (4 x^3+e^{3 c_1}\right ){}^3}-2 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}{4 \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 -x^2+\frac {\left (1+i \sqrt {3}\right ) x \left (-x^3+2 e^{3 c_1}\right )}{4 \sqrt [3]{x^6+5 e^{3 c_1} x^3+\frac {1}{2} \sqrt {e^{3 c_1} \left (4 x^3+e^{3 c_1}\right ){}^3}-\frac {e^{6 c_1}}{2}}}+\frac {1}{4} i \left (\sqrt {3}+i\right ) \sqrt [3]{x^6+5 e^{3 c_1} x^3+\frac {1}{2} \sqrt {e^{3 c_1} \left (4 x^3+e^{3 c_1}\right ){}^3}-\frac {e^{6 c_1}}{2}}\\ y(x)&\to -x^2+\frac {i \left (\sqrt {3}+i\right ) x \left (x^3-2 e^{3 c_1}\right )}{4 \sqrt [3]{x^6+5 e^{3 c_1} x^3+\frac {1}{2} \sqrt {e^{3 c_1} \left (4 x^3+e^{3 c_1}\right ){}^3}-\frac {e^{6 c_1}}{2}}}-\frac {1}{4} \left (1+i \sqrt {3}\right ) \sqrt [3]{x^6+5 e^{3 c_1} x^3+\frac {1}{2} \sqrt {e^{3 c_1} \left (4 x^3+e^{3 c_1}\right ){}^3}-\frac {e^{6 c_1}}{2}} \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(2*x*Derivative(y(x), x) + 2*y(x) + Derivative(y(x), x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE x - sqrt(x**2 - 2*y(x)) + Derivative(y(x), x) cannot be solved by the factorable group method