71.1.102 problem 129 (page 179)
Internal
problem
ID
[19278]
Book
:
V.V.
Stepanov,
A
course
of
differential
equations
(in
Russian),
GIFML.
Moscow
(1958)
Section
:
All
content
Problem
number
:
129
(page
179)
Date
solved
:
Thursday, October 02, 2025 at 04:18:18 PM
CAS
classification
:
[[_2nd_order, _missing_y]]
\begin{align*} {y^{\prime \prime }}^{2}-2 x y^{\prime \prime }-y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.151 (sec). Leaf size: 205
ode:=diff(diff(y(x),x),x)^2-2*x*diff(diff(y(x),x),x)-diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \int \left (-x^{2}+\operatorname {RootOf}\left (8 \textit {\_Z}^{5}+20 \textit {\_Z}^{4} x -10 \textit {\_Z}^{3} x^{2}-45 \textit {\_Z}^{2} x^{3}+27 x^{5}-c_1 \right )^{2}\right )d x +c_2 \\
y &= \int \left (-x^{2}+\operatorname {RootOf}\left (8 \textit {\_Z}^{5}+20 \textit {\_Z}^{4} x -10 \textit {\_Z}^{3} x^{2}-45 \textit {\_Z}^{2} x^{3}+27 x^{5}+c_1 \right )^{2}\right )d x +c_2 \\
y &= \int \left (-x^{2}+\operatorname {RootOf}\left (8 \textit {\_Z}^{5}-20 \textit {\_Z}^{4} x -10 \textit {\_Z}^{3} x^{2}+45 \textit {\_Z}^{2} x^{3}-27 x^{5}-c_1 \right )^{2}\right )d x +c_2 \\
y &= \int \left (-x^{2}+\operatorname {RootOf}\left (8 \textit {\_Z}^{5}-20 \textit {\_Z}^{4} x -10 \textit {\_Z}^{3} x^{2}+45 \textit {\_Z}^{2} x^{3}-27 x^{5}+c_1 \right )^{2}\right )d x +c_2 \\
\end{align*}
✓ Mathematica. Time used: 0.115 (sec). Leaf size: 501
ode=D[y[x],{x,2}]^2-2*x*D[y[x],{x,2}]-D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \int _1^x\text {Root}\left [4 e^{5 c_1} K[1]^5+300 \text {$\#$1}^3 K[1]^4-10 e^{5 c_1} \text {$\#$1} K[1]^3-240 \text {$\#$1}^4 K[1]^2-e^{10 c_1}+64 \text {$\#$1}^5+\left (40 e^{5 c_1} K[1]-125 K[1]^6\right ) \text {$\#$1}^2\&,1\right ]dK[1]+c_2\\ y(x)&\to \int _1^x\text {Root}\left [4 e^{5 c_1} K[2]^5+300 \text {$\#$1}^3 K[2]^4-10 e^{5 c_1} \text {$\#$1} K[2]^3-240 \text {$\#$1}^4 K[2]^2-e^{10 c_1}+64 \text {$\#$1}^5+\left (40 e^{5 c_1} K[2]-125 K[2]^6\right ) \text {$\#$1}^2\&,2\right ]dK[2]+c_2\\ y(x)&\to \int _1^x\text {Root}\left [4 e^{5 c_1} K[3]^5+300 \text {$\#$1}^3 K[3]^4-10 e^{5 c_1} \text {$\#$1} K[3]^3-240 \text {$\#$1}^4 K[3]^2-e^{10 c_1}+64 \text {$\#$1}^5+\left (40 e^{5 c_1} K[3]-125 K[3]^6\right ) \text {$\#$1}^2\&,3\right ]dK[3]+c_2\\ y(x)&\to \int _1^x\text {Root}\left [4 e^{5 c_1} K[4]^5+300 \text {$\#$1}^3 K[4]^4-10 e^{5 c_1} \text {$\#$1} K[4]^3-240 \text {$\#$1}^4 K[4]^2-e^{10 c_1}+64 \text {$\#$1}^5+\left (40 e^{5 c_1} K[4]-125 K[4]^6\right ) \text {$\#$1}^2\&,4\right ]dK[4]+c_2\\ y(x)&\to \int _1^x\text {Root}\left [4 e^{5 c_1} K[5]^5+300 \text {$\#$1}^3 K[5]^4-10 e^{5 c_1} \text {$\#$1} K[5]^3-240 \text {$\#$1}^4 K[5]^2-e^{10 c_1}+64 \text {$\#$1}^5+\left (40 e^{5 c_1} K[5]-125 K[5]^6\right ) \text {$\#$1}^2\&,5\right ]dK[5]+c_2 \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-2*x*Derivative(y(x), (x, 2)) - Derivative(y(x), x) + Derivative(y(x), (x, 2))**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -(-2*x + Derivative(y(x), (x, 2)))*Derivative(y(x), (x, 2)) + Derivative(y(x), x) cannot be solved by the factorable group method