23.4.284 problem 287
Internal
problem
ID
[6586]
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
:
287
Date
solved
:
Tuesday, September 30, 2025 at 03:16:44 PM
CAS
classification
:
[[_2nd_order, _missing_y], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_poly_yn]]
\begin{align*} \left (x -{y^{\prime }}^{2}\right ) y^{\prime \prime }&=x^{2}-y^{\prime } \end{align*}
✓ Maple. Time used: 0.013 (sec). Leaf size: 329
ode:=(x-diff(y(x),x)^2)*diff(diff(y(x),x),x) = x^2-diff(y(x),x);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\int \frac {\left (-4 x^{3}+12 c_1 +4 \sqrt {x^{6}+\left (-6 c_1 -4\right ) x^{3}+9 c_1^{2}}\right )^{{2}/{3}}+4 x}{\left (-4 x^{3}+12 c_1 +4 \sqrt {x^{6}+\left (-6 c_1 -4\right ) x^{3}+9 c_1^{2}}\right )^{{1}/{3}}}d x}{2}+c_2 \\
y &= -\frac {\int \frac {i \left (\left (-4 x^{3}+12 c_1 +4 \sqrt {x^{6}+\left (-6 c_1 -4\right ) x^{3}+9 c_1^{2}}\right )^{{2}/{3}}-4 x \right ) \sqrt {3}+\left (-4 x^{3}+12 c_1 +4 \sqrt {x^{6}+\left (-6 c_1 -4\right ) x^{3}+9 c_1^{2}}\right )^{{2}/{3}}+4 x}{\left (-4 x^{3}+12 c_1 +4 \sqrt {x^{6}+\left (-6 c_1 -4\right ) x^{3}+9 c_1^{2}}\right )^{{1}/{3}}}d x}{4}+c_2 \\
y &= \frac {\int \frac {i \left (\left (-4 x^{3}+12 c_1 +4 \sqrt {x^{6}+\left (-6 c_1 -4\right ) x^{3}+9 c_1^{2}}\right )^{{2}/{3}}-4 x \right ) \sqrt {3}-\left (-4 x^{3}+12 c_1 +4 \sqrt {x^{6}+\left (-6 c_1 -4\right ) x^{3}+9 c_1^{2}}\right )^{{2}/{3}}-4 x}{\left (-4 x^{3}+12 c_1 +4 \sqrt {x^{6}+\left (-6 c_1 -4\right ) x^{3}+9 c_1^{2}}\right )^{{1}/{3}}}d x}{4}+c_2 \\
\end{align*}
✓ Mathematica. Time used: 62.432 (sec). Leaf size: 394
ode=(x - D[y[x],x]^2)*D[y[x],{x,2}] == x^2 - D[y[x],x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \int _1^x\left (-\frac {\sqrt [3]{2} K[1]}{\sqrt [3]{K[1]^3+3 c_1+\sqrt {K[1]^6+6 c_1 K[1]^3-4 K[1]^3+9 c_1{}^2}}}-\frac {\sqrt [3]{K[1]^3+3 c_1+\sqrt {K[1]^6+6 c_1 K[1]^3-4 K[1]^3+9 c_1{}^2}}}{\sqrt [3]{2}}\right )dK[1]+c_2\\ y(x)&\to \int _1^x\frac {\sqrt [3]{2} \left (2+2 i \sqrt {3}\right ) K[2]+2^{2/3} \left (1-i \sqrt {3}\right ) \left (K[2]^3+3 c_1+\sqrt {K[2]^6+6 c_1 K[2]^3-4 K[2]^3+9 c_1{}^2}\right ){}^{2/3}}{4 \sqrt [3]{K[2]^3+3 c_1+\sqrt {K[2]^6+6 c_1 K[2]^3-4 K[2]^3+9 c_1{}^2}}}dK[2]+c_2\\ y(x)&\to \int _1^x\frac {\sqrt [3]{2} \left (2-2 i \sqrt {3}\right ) K[3]+2^{2/3} \left (1+i \sqrt {3}\right ) \left (K[3]^3+3 c_1+\sqrt {K[3]^6+6 c_1 K[3]^3-4 K[3]^3+9 c_1{}^2}\right ){}^{2/3}}{4 \sqrt [3]{K[3]^3+3 c_1+\sqrt {K[3]^6+6 c_1 K[3]^3-4 K[3]^3+9 c_1{}^2}}}dK[3]+c_2 \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x**2 + (x - Derivative(y(x), x)**2)*Derivative(y(x), (x, 2)) + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out