23.4.262 problem 262
Internal
problem
ID
[6564]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Part
II.
Chapter
4.
THE
NONLINEAR
EQUATION
OF
SECOND
ORDER,
page
380
Problem
number
:
262
Date
solved
:
Tuesday, September 30, 2025 at 03:04:32 PM
CAS
classification
:
[[_2nd_order, _missing_x]]
\begin{align*} y^{2} {y^{\prime }}^{2}+2 y^{3} y^{\prime \prime }&=2 \end{align*}
✓ Maple. Time used: 0.015 (sec). Leaf size: 767
ode:=y(x)^2*diff(y(x),x)^2+2*y(x)^3*diff(diff(y(x),x),x) = 2;
dsolve(ode,y(x), singsol=all);
\begin{align*} \text {Solution too large to show}\end{align*}
✓ Mathematica. Time used: 60.25 (sec). Leaf size: 676
ode=y[x]^2*D[y[x],x]^2 + 2*y[x]^3*D[y[x],{x,2}] == 2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {c_1{}^2 \left (3 \sqrt {\frac {(x+c_2){}^2 \left (9 c_1{}^4 x^2+18 c_2 c_1{}^4 x+128+9 c_2{}^2 c_1{}^4\right )}{c_1{}^2}}+9 c_1 (x+c_2){}^2+\frac {64}{c_1{}^3}\right ){}^{2/3}-4 c_1 \sqrt [3]{3 \sqrt {\frac {(x+c_2){}^2 \left (9 c_1{}^4 x^2+18 c_2 c_1{}^4 x+128+9 c_2{}^2 c_1{}^4\right )}{c_1{}^2}}+9 c_1 (x+c_2){}^2+\frac {64}{c_1{}^3}}+16}{2 c_1{}^2 \sqrt [3]{3 \sqrt {\frac {(x+c_2){}^2 \left (9 c_1{}^4 x^2+18 c_2 c_1{}^4 x+128+9 c_2{}^2 c_1{}^4\right )}{c_1{}^2}}+9 c_1 (x+c_2){}^2+\frac {64}{c_1{}^3}}}\\ y(x)&\to \frac {i \left (4+c_1 \sqrt [3]{3 \sqrt {\frac {(x+c_2){}^2 \left (9 c_1{}^4 x^2+18 c_2 c_1{}^4 x+128+9 c_2{}^2 c_1{}^4\right )}{c_1{}^2}}+9 c_1 (x+c_2){}^2+\frac {64}{c_1{}^3}}\right ) \left (\left (\sqrt {3}+i\right ) c_1 \sqrt [3]{3 \sqrt {\frac {(x+c_2){}^2 \left (9 c_1{}^4 x^2+18 c_2 c_1{}^4 x+128+9 c_2{}^2 c_1{}^4\right )}{c_1{}^2}}+9 c_1 (x+c_2){}^2+\frac {64}{c_1{}^3}}-4 \sqrt {3}+4 i\right )}{4 c_1{}^2 \sqrt [3]{3 \sqrt {\frac {(x+c_2){}^2 \left (9 c_1{}^4 x^2+18 c_2 c_1{}^4 x+128+9 c_2{}^2 c_1{}^4\right )}{c_1{}^2}}+9 c_1 (x+c_2){}^2+\frac {64}{c_1{}^3}}}\\ y(x)&\to -\frac {1}{4} i \left (\left (\sqrt {3}-i\right ) \sqrt [3]{3 \sqrt {\frac {(x+c_2){}^2 \left (9 c_1{}^4 x^2+18 c_2 c_1{}^4 x+128+9 c_2{}^2 c_1{}^4\right )}{c_1{}^2}}+9 c_1 (x+c_2){}^2+\frac {64}{c_1{}^3}}-\frac {16 \left (\sqrt {3}+i\right )}{c_1{}^2 \sqrt [3]{3 \sqrt {\frac {(x+c_2){}^2 \left (9 c_1{}^4 x^2+18 c_2 c_1{}^4 x+128+9 c_2{}^2 c_1{}^4\right )}{c_1{}^2}}+9 c_1 (x+c_2){}^2+\frac {64}{c_1{}^3}}}-\frac {4 \left (\sqrt {3}+i\right )}{c_1}+\frac {4 \left (\sqrt {3}-i\right )}{c_1}\right ) \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(2*y(x)**3*Derivative(y(x), (x, 2)) + y(x)**2*Derivative(y(x), x)**2 - 2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -sqrt(-2*y(x)**3*Derivative(y(x), (x, 2)) + 2)/y(x) + Derivative