77.1.80 problem 106 (page 162)
Internal
problem
ID
[17891]
Book
:
V.V.
Stepanov,
A
course
of
differential
equations
(in
Russian),
GIFML.
Moscow
(1958)
Section
:
All
content
Problem
number
:
106
(page
162)
Date
solved
:
Thursday, March 13, 2025 at 11:09:03 AM
CAS
classification
:
[[_3rd_order, _quadrature]]
\begin{align*} {y^{\prime \prime \prime }}^{2}+x^{2}&=1 \end{align*}
✓ Maple. Time used: 0.091 (sec). Leaf size: 91
ode:=diff(diff(diff(y(x),x),x),x)^2+x^2 = 1;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (2 x^{3}+13 x \right ) \sqrt {-x^{2}+1}}{48}+\frac {\left (4 x^{2}+1\right ) \arcsin \left (x \right )}{16}+\frac {c_{1} x^{2}}{2}+c_{2} x +c_{3} \\
y &= \frac {\left (-2 x^{3}-13 x \right ) \sqrt {-x^{2}+1}}{48}+\frac {\left (-4 x^{2}-1\right ) \arcsin \left (x \right )}{16}+\frac {c_{1} x^{2}}{2}+c_{2} x +c_{3} \\
\end{align*}
✓ Mathematica. Time used: 0.114 (sec). Leaf size: 147
ode=D[y[x],{x,3}]^2+x^2==1;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {1}{48} \left (6 \left (2 x^2-1\right ) \arcsin (x)-18 \arctan \left (\frac {\sqrt {1-x^2}}{x+1}\right )+x \sqrt {1-x^2} \left (2 x^2+13\right )\right )+c_3 x^2+c_2 x+c_1 \\
y(x)\to \frac {1}{48} \left (\left (6-12 x^2\right ) \arcsin (x)+18 \arctan \left (\frac {\sqrt {1-x^2}}{x+1}\right )-x \sqrt {1-x^2} \left (2 x^2+13\right )\right )+c_3 x^2+c_2 x+c_1 \\
\end{align*}
✓ Sympy. Time used: 22.006 (sec). Leaf size: 286
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x**2 + Derivative(y(x), (x, 3))**2 - 1,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = C_{1} + C_{2} x + C_{3} x^{2} - \frac {x^{2} \operatorname {asin}{\left (x \right )}}{4} - \frac {7 x \sqrt {1 - x^{2}}}{24} - \frac {\begin {cases} \frac {i x^{5}}{4 \sqrt {x^{2} - 1}} - \frac {3 i x^{3}}{8 \sqrt {x^{2} - 1}} + \frac {i x}{8 \sqrt {x^{2} - 1}} - \frac {i \operatorname {acosh}{\left (x \right )}}{8} & \text {for}\: \left |{x^{2}}\right | > 1 \\- \frac {x^{5}}{4 \sqrt {1 - x^{2}}} + \frac {3 x^{3}}{8 \sqrt {1 - x^{2}}} - \frac {x}{8 \sqrt {1 - x^{2}}} + \frac {\operatorname {asin}{\left (x \right )}}{8} & \text {otherwise} \end {cases}}{6} - \frac {\operatorname {asin}{\left (x \right )}}{24}, \ y{\left (x \right )} = C_{1} + C_{2} x + C_{3} x^{2} + \frac {x^{2} \operatorname {asin}{\left (x \right )}}{4} + \frac {7 x \sqrt {1 - x^{2}}}{24} + \frac {\begin {cases} \frac {i x^{5}}{4 \sqrt {x^{2} - 1}} - \frac {3 i x^{3}}{8 \sqrt {x^{2} - 1}} + \frac {i x}{8 \sqrt {x^{2} - 1}} - \frac {i \operatorname {acosh}{\left (x \right )}}{8} & \text {for}\: \left |{x^{2}}\right | > 1 \\- \frac {x^{5}}{4 \sqrt {1 - x^{2}}} + \frac {3 x^{3}}{8 \sqrt {1 - x^{2}}} - \frac {x}{8 \sqrt {1 - x^{2}}} + \frac {\operatorname {asin}{\left (x \right )}}{8} & \text {otherwise} \end {cases}}{6} + \frac {\operatorname {asin}{\left (x \right )}}{24}\right ]
\]