71.1.82 problem 108 (page 162)
Internal
problem
ID
[19258]
Book
:
V.V.
Stepanov,
A
course
of
differential
equations
(in
Russian),
GIFML.
Moscow
(1958)
Section
:
All
content
Problem
number
:
108
(page
162)
Date
solved
:
Thursday, October 02, 2025 at 04:03:35 PM
CAS
classification
:
[[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2]]
\begin{align*} a^{3} y^{\prime \prime \prime } y^{\prime \prime }&=\sqrt {1+c^{2} {y^{\prime \prime }}^{2}} \end{align*}
✓ Maple. Time used: 0.096 (sec). Leaf size: 223
ode:=a^3*diff(diff(diff(y(x),x),x),x)*diff(diff(y(x),x),x) = (1+c^2*diff(diff(y(x),x),x)^2)^(1/2);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -\frac {i x^{2}}{2 c}+c_1 x +c_2 \\
y &= \frac {i x^{2}}{2 c}+c_1 x +c_2 \\
y &= \frac {-\int \left (a^{6} \ln \left (\sqrt {\left (c_1 +x \right )^{2} c^{4}-a^{6}}+\left (c_1 +x \right ) c^{2} \operatorname {csgn}\left (c^{2}\right )\right ) \operatorname {csgn}\left (c^{2}\right )-c^{2} \sqrt {\left (c_1 +x \right )^{2} c^{4}-a^{6}}\, \left (c_1 +x \right )\right )d x +\left (2 c_2 x +2 c_3 \right ) c^{3} a^{3}}{2 c^{3} a^{3}} \\
y &= \frac {-\int \left (-a^{6} \ln \left (\sqrt {\left (c_1 +x \right )^{2} c^{4}-a^{6}}+\left (c_1 +x \right ) c^{2} \operatorname {csgn}\left (c^{2}\right )\right ) \operatorname {csgn}\left (c^{2}\right )+c^{2} \sqrt {\left (c_1 +x \right )^{2} c^{4}-a^{6}}\, \left (c_1 +x \right )\right )d x +\left (2 c_2 x +2 c_3 \right ) c^{3} a^{3}}{2 c^{3} a^{3}} \\
\end{align*}
✓ Mathematica. Time used: 22.278 (sec). Leaf size: 438
ode=a^3*D[y[x],{x,3}]*D[y[x],{x,2}]== Sqrt[1+c^2* D[y[x],{x,2}]^2];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {\sqrt {a^6 \left (-1+c^4 c_1{}^2\right )+2 a^3 c^4 c_1 x+c^4 x^2} \left (a^6 \left (2+c^4 c_1{}^2\right )+2 a^3 c^4 c_1 x+c^4 x^2\right )-3 a^6 c^2 x \log \left (c^2 \left (c^2 \left (x+a^3 c_1\right )+\sqrt {a^6 \left (-1+c^4 c_1{}^2\right )+2 a^3 c^4 c_1 x+c^4 x^2}\right )\right )-3 a^9 c^2 c_1 \log \left (c^2 \left (x+a^3 c_1\right )+\sqrt {a^6 \left (-1+c^4 c_1{}^2\right )+2 a^3 c^4 c_1 x+c^4 x^2}\right )}{6 a^3 c^5}+c_3 x+c_2\\ y(x)&\to \frac {-\sqrt {a^6 \left (-1+c^4 c_1{}^2\right )+2 a^3 c^4 c_1 x+c^4 x^2} \left (a^6 \left (2+c^4 c_1{}^2\right )+2 a^3 c^4 c_1 x+c^4 x^2\right )+3 a^6 c^2 x \log \left (c^2 \left (c^2 \left (x+a^3 c_1\right )+\sqrt {a^6 \left (-1+c^4 c_1{}^2\right )+2 a^3 c^4 c_1 x+c^4 x^2}\right )\right )+3 a^9 c^2 c_1 \log \left (c^2 \left (x+a^3 c_1\right )+\sqrt {a^6 \left (-1+c^4 c_1{}^2\right )+2 a^3 c^4 c_1 x+c^4 x^2}\right )}{6 a^3 c^5}+c_3 x+c_2 \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
c = symbols("c")
y = Function("y")
ode = Eq(a**3*Derivative(y(x), (x, 2))*Derivative(y(x), (x, 3)) - sqrt(c**2*Derivative(y(x), (x, 2))**2 + 1),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
ODEMatchError : nth_linear_constant_coeff_undetermined_coefficients solver cannot solve:
nan