23.6.9 problem 9
Internal
problem
ID
[6808]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Part
II.
Chapter
6.
THE
EQUATION
IS
NONLINEAR
AND
OF
ORDER
GREATER
THAN
TWO,
page
427
Problem
number
:
9
Date
solved
:
Friday, October 03, 2025 at 02:09:54 AM
CAS
classification
:
[[_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries]]
\begin{align*} 2 {y^{\prime }}^{3}+3 y^{\prime \prime }+6 y y^{\prime } y^{\prime \prime }+\left (x +y^{2}\right ) y^{\prime \prime \prime }&=0 \end{align*}
✓ Maple. Time used: 0.014 (sec). Leaf size: 584
ode:=2*diff(y(x),x)^3+3*diff(diff(y(x),x),x)+6*y(x)*diff(y(x),x)*diff(diff(y(x),x),x)+(x+y(x)^2)*diff(diff(diff(y(x),x),x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (-6 c_1 \,x^{2}-12 c_2 x +12 c_3 +2 \sqrt {9 c_1^{2} x^{4}+36 c_1 c_2 \,x^{3}-36 c_1 c_3 \,x^{2}+36 c_2^{2} x^{2}-72 c_2 c_3 x +16 x^{3}+36 c_3^{2}}\right )^{{2}/{3}}-4 x}{2 \left (-6 c_1 \,x^{2}-12 c_2 x +12 c_3 +2 \sqrt {9 c_1^{2} x^{4}+36 c_1 c_2 \,x^{3}-36 c_1 c_3 \,x^{2}+36 c_2^{2} x^{2}-72 c_2 c_3 x +16 x^{3}+36 c_3^{2}}\right )^{{1}/{3}}} \\
y &= -\frac {i \sqrt {3}\, \left (-6 c_1 \,x^{2}-12 c_2 x +12 c_3 +2 \sqrt {9 c_1^{2} x^{4}+36 c_1 c_2 \,x^{3}-36 c_1 c_3 \,x^{2}+36 c_2^{2} x^{2}-72 c_2 c_3 x +16 x^{3}+36 c_3^{2}}\right )^{{2}/{3}}+4 i \sqrt {3}\, x +\left (-6 c_1 \,x^{2}-12 c_2 x +12 c_3 +2 \sqrt {9 c_1^{2} x^{4}+36 c_1 c_2 \,x^{3}-36 c_1 c_3 \,x^{2}+36 c_2^{2} x^{2}-72 c_2 c_3 x +16 x^{3}+36 c_3^{2}}\right )^{{2}/{3}}-4 x}{4 \left (-6 c_1 \,x^{2}-12 c_2 x +12 c_3 +2 \sqrt {9 c_1^{2} x^{4}+36 c_1 c_2 \,x^{3}-36 c_1 c_3 \,x^{2}+36 c_2^{2} x^{2}-72 c_2 c_3 x +16 x^{3}+36 c_3^{2}}\right )^{{1}/{3}}} \\
y &= \frac {i \sqrt {3}\, \left (-6 c_1 \,x^{2}-12 c_2 x +12 c_3 +2 \sqrt {9 c_1^{2} x^{4}+36 c_1 c_2 \,x^{3}-36 c_1 c_3 \,x^{2}+36 c_2^{2} x^{2}-72 c_2 c_3 x +16 x^{3}+36 c_3^{2}}\right )^{{2}/{3}}+4 i \sqrt {3}\, x -\left (-6 c_1 \,x^{2}-12 c_2 x +12 c_3 +2 \sqrt {9 c_1^{2} x^{4}+36 c_1 c_2 \,x^{3}-36 c_1 c_3 \,x^{2}+36 c_2^{2} x^{2}-72 c_2 c_3 x +16 x^{3}+36 c_3^{2}}\right )^{{2}/{3}}+4 x}{4 \left (-6 c_1 \,x^{2}-12 c_2 x +12 c_3 +2 \sqrt {9 c_1^{2} x^{4}+36 c_1 c_2 \,x^{3}-36 c_1 c_3 \,x^{2}+36 c_2^{2} x^{2}-72 c_2 c_3 x +16 x^{3}+36 c_3^{2}}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 0.151 (sec). Leaf size: 457
ode=2*D[y[x],x]^3 + 3*D[y[x],{x,2}] + 6*y[x]*D[y[x],x]*D[y[x],{x,2}] + (x + y[x]^2)*D[y[x],{x,3}] == 0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {-2\ 2^{2/3} x+\sqrt [3]{2} \left (3 c_1 x^2+\sqrt {16 x^3+\left (3 c_1 x^2+c_3 x+6 c_2\right ){}^2}+c_3 x+6 c_2\right ){}^{2/3}}{2 \sqrt [3]{3 c_1 x^2+\sqrt {16 x^3+\left (3 c_1 x^2+c_3 x+6 c_2\right ){}^2}+c_3 x+6 c_2}}\\ y(x)&\to \frac {i \sqrt [3]{2} \left (\sqrt {3}+i\right ) \left (3 c_1 x^2+\sqrt {9 c_1{}^2 x^4+2 (8+3 c_1 c_3) x^3+\left (c_3{}^2+36 c_1 c_2\right ) x^2+12 c_2 c_3 x+36 c_2{}^2}+c_3 x+6 c_2\right ){}^{2/3}+2\ 2^{2/3} \left (1+i \sqrt {3}\right ) x}{4 \sqrt [3]{3 c_1 x^2+\sqrt {16 x^3+\left (3 c_1 x^2+c_3 x+6 c_2\right ){}^2}+c_3 x+6 c_2}}\\ y(x)&\to \frac {\sqrt [3]{2} \left (-1-i \sqrt {3}\right ) \left (3 c_1 x^2+\sqrt {9 c_1{}^2 x^4+2 (8+3 c_1 c_3) x^3+\left (c_3{}^2+36 c_1 c_2\right ) x^2+12 c_2 c_3 x+36 c_2{}^2}+c_3 x+6 c_2\right ){}^{2/3}+2\ 2^{2/3} \left (1-i \sqrt {3}\right ) x}{4 \sqrt [3]{3 c_1 x^2+\sqrt {16 x^3+\left (3 c_1 x^2+c_3 x+6 c_2\right ){}^2}+c_3 x+6 c_2}} \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq((x + y(x)**2)*Derivative(y(x), (x, 3)) + 6*y(x)*Derivative(y(x), x)*Derivative(y(x), (x, 2)) + 2*Derivative(y(x), x)**3 + 3*Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE (27*x*Derivative(y(x), (x, 3))/4 + sqrt((27*x*Derivative(y(x), (