60.7.7 problem 1597 (6.7)
Internal
problem
ID
[11557]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
6,
non-linear
second
order
Problem
number
:
1597
(6.7)
Date
solved
:
Wednesday, March 05, 2025 at 02:28:00 PM
CAS
classification
:
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\begin{align*} y^{\prime \prime }-a y^{3}&=0 \end{align*}
✓ Maple. Time used: 0.018 (sec). Leaf size: 25
ode:=diff(diff(y(x),x),x)-a*y(x)^3 = 0;
dsolve(ode,y(x), singsol=all);
\[
y = c_{2} \operatorname {JacobiSN}\left (\frac {\left (\sqrt {2}\, \sqrt {-a}\, x +2 c_{1} \right ) c_{2}}{2}, i\right )
\]
✓ Mathematica. Time used: 61.202 (sec). Leaf size: 131
ode=-(a*y[x]^3) + D[y[x],{x,2}] == 0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {i \sqrt [4]{2} \text {sn}\left (\left .-\frac {(1-i) \sqrt {\sqrt {a} \sqrt {c_1} (x+c_2){}^2}}{2^{3/4}}\right |-1\right )}{\sqrt {\frac {i \sqrt {a}}{\sqrt {c_1}}}} \\
y(x)\to \frac {i \sqrt [4]{2} \text {sn}\left (\left .-\frac {(1-i) \sqrt {\sqrt {a} \sqrt {c_1} (x+c_2){}^2}}{2^{3/4}}\right |-1\right )}{\sqrt {\frac {i \sqrt {a}}{\sqrt {c_1}}}} \\
\end{align*}
✓ Sympy. Time used: 14.741 (sec). Leaf size: 88
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(-a*y(x)**3 + Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ \frac {y{\left (x \right )} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {a e^{i \pi } y^{4}{\left (x \right )}}{2 C_{1}}} \right )}}{4 \sqrt {C_{1}} \Gamma \left (\frac {5}{4}\right )} = C_{2} + x, \ \frac {y{\left (x \right )} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {a e^{i \pi } y^{4}{\left (x \right )}}{2 C_{1}}} \right )}}{4 \sqrt {C_{1}} \Gamma \left (\frac {5}{4}\right )} = C_{2} - x\right ]
\]