23.6.19 problem 19
Internal
problem
ID
[6818]
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
:
19
Date
solved
:
Tuesday, September 30, 2025 at 03:52:26 PM
CAS
classification
:
[[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2]]
\begin{align*} y^{\prime \prime } y^{\prime \prime \prime }&=a \sqrt {1+b^{2} {y^{\prime \prime }}^{2}} \end{align*}
✓ Maple. Time used: 0.128 (sec). Leaf size: 295
ode:=diff(diff(y(x),x),x)*diff(diff(diff(y(x),x),x),x) = a*(1+b^2*diff(diff(y(x),x),x)^2)^(1/2);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -\frac {i x^{2}}{2 b}+c_1 x +c_2 \\
y &= \frac {i x^{2}}{2 b}+c_1 x +c_2 \\
y &= -\frac {\int \left (-\left (c_1 +x \right ) \sqrt {a^{2} b^{4}}\, \sqrt {\left (-1+b^{2} \left (c_1 +x \right ) a \right ) \left (1+b^{2} \left (c_1 +x \right ) a \right )}+\ln \left (\frac {a^{2} b^{4} \left (c_1 +x \right )+\sqrt {\left (-1+b^{2} \left (c_1 +x \right ) a \right ) \left (1+b^{2} \left (c_1 +x \right ) a \right )}\, \sqrt {a^{2} b^{4}}}{\sqrt {a^{2} b^{4}}}\right )\right )d x -2 b \sqrt {a^{2} b^{4}}\, \left (c_2 x +c_3 \right )}{2 \sqrt {a^{2} b^{4}}\, b} \\
y &= \frac {\int \left (-\left (c_1 +x \right ) \sqrt {a^{2} b^{4}}\, \sqrt {\left (-1+b^{2} \left (c_1 +x \right ) a \right ) \left (1+b^{2} \left (c_1 +x \right ) a \right )}+\ln \left (\frac {a^{2} b^{4} \left (c_1 +x \right )+\sqrt {\left (-1+b^{2} \left (c_1 +x \right ) a \right ) \left (1+b^{2} \left (c_1 +x \right ) a \right )}\, \sqrt {a^{2} b^{4}}}{\sqrt {a^{2} b^{4}}}\right )\right )d x +2 b \sqrt {a^{2} b^{4}}\, \left (c_2 x +c_3 \right )}{2 \sqrt {a^{2} b^{4}}\, b} \\
\end{align*}
✓ Mathematica. Time used: 18.266 (sec). Leaf size: 415
ode=D[y[x],{x,2}]*D[y[x],{x,3}] == a*Sqrt[1 + b^2*D[y[x],{x,2}]^2];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {6 a^2 b^5 c_3 x+6 a^2 b^5 c_2+\left (a^2 b^4 x^2+2 a b^4 c_1 x+b^4 c_1{}^2-1\right ){}^{3/2}+3 \sqrt {a^2 b^4 x^2+2 a b^4 c_1 x+b^4 c_1{}^2-1}-3 b^2 c_1 \log \left (\sqrt {a^2 b^4 x^2+2 a b^4 c_1 x+b^4 c_1{}^2-1}+a b^2 x+b^2 c_1\right )-3 a b^2 x \log \left (b^2 \left (\sqrt {a^2 b^4 x^2+2 a b^4 c_1 x+b^4 c_1{}^2-1}+a b^2 x+b^2 c_1\right )\right )}{6 a^2 b^5}\\ y(x)&\to \frac {-\sqrt {a^2 b^4 x^2+2 a b^4 c_1 x+b^4 c_1{}^2-1} \left (a^2 b^4 x^2+2 a b^4 c_1 x+b^4 c_1{}^2+2\right )+3 b^2 c_1 \log \left (\sqrt {a^2 b^4 x^2+2 a b^4 c_1 x+b^4 c_1{}^2-1}+a b^2 x+b^2 c_1\right )+3 a b^2 x \log \left (b^2 \left (\sqrt {a^2 b^4 x^2+2 a b^4 c_1 x+b^4 c_1{}^2-1}+a b^2 x+b^2 c_1\right )\right )}{6 a^2 b^5}+c_3 x+c_2 \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq(-a*sqrt(b**2*Derivative(y(x), (x, 2))**2 + 1) + Derivative(y(x), (x, 2))*Derivative(y(x), (x, 3)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
ODEMatchError : nth_linear_constant_coeff_undetermined_coefficients solver cannot solve:
nan