60.8.12 problem 1848
Internal
problem
ID
[11773]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
7,
non-linear
third
and
higher
order
Problem
number
:
1848
Date
solved
:
Wednesday, March 05, 2025 at 03:06:34 PM
CAS
classification
:
[[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2], [_3rd_order, _reducible, _mu_poly_yn]]
\begin{align*} \left ({y^{\prime }}^{2}+1\right ) y^{\prime \prime \prime }-\left (3 y^{\prime }+a \right ) {y^{\prime \prime }}^{2}&=0 \end{align*}
✓ Maple. Time used: 0.215 (sec). Leaf size: 375
ode:=(1+diff(y(x),x)^2)*diff(diff(diff(y(x),x),x),x)-(3*diff(y(x),x)+a)*diff(diff(y(x),x),x)^2 = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -i x +c_{1} \\
y &= i x +c_{1} \\
y &= \int \tan \left (\operatorname {RootOf}\left (-2 \cos \left (\textit {\_Z} \right ) {\mathrm e}^{a \textit {\_Z}} c_{1} c_{2} a^{3}-2 \cos \left (\textit {\_Z} \right ) {\mathrm e}^{a \textit {\_Z}} c_{1} a^{3} x +c_{2}^{2} a^{4} {\mathrm e}^{2 a \textit {\_Z}}+2 c_{2} a^{4} x \,{\mathrm e}^{2 a \textit {\_Z}}+a^{4} x^{2} {\mathrm e}^{2 a \textit {\_Z}}+\cos \left (\textit {\_Z} \right )^{2} c_{1}^{2} a^{2}-2 \cos \left (\textit {\_Z} \right ) {\mathrm e}^{a \textit {\_Z}} c_{1} c_{2} a -2 \cos \left (\textit {\_Z} \right ) {\mathrm e}^{a \textit {\_Z}} c_{1} a x +2 c_{2}^{2} a^{2} {\mathrm e}^{2 a \textit {\_Z}}+4 c_{2} a^{2} x \,{\mathrm e}^{2 a \textit {\_Z}}+2 a^{2} x^{2} {\mathrm e}^{2 a \textit {\_Z}}-\sin \left (\textit {\_Z} \right )^{2} c_{1}^{2}+c_{2}^{2} {\mathrm e}^{2 a \textit {\_Z}}+2 c_{2} x \,{\mathrm e}^{2 a \textit {\_Z}}+x^{2} {\mathrm e}^{2 a \textit {\_Z}}\right )\right )d x +c_3 \\
y &= \int \tan \left (\operatorname {RootOf}\left (2 \cos \left (\textit {\_Z} \right ) {\mathrm e}^{a \textit {\_Z}} c_{1} c_{2} a^{3}+2 \cos \left (\textit {\_Z} \right ) {\mathrm e}^{a \textit {\_Z}} c_{1} a^{3} x +c_{2}^{2} a^{4} {\mathrm e}^{2 a \textit {\_Z}}+2 c_{2} a^{4} x \,{\mathrm e}^{2 a \textit {\_Z}}+a^{4} x^{2} {\mathrm e}^{2 a \textit {\_Z}}+\cos \left (\textit {\_Z} \right )^{2} c_{1}^{2} a^{2}+2 \cos \left (\textit {\_Z} \right ) {\mathrm e}^{a \textit {\_Z}} c_{1} c_{2} a +2 \cos \left (\textit {\_Z} \right ) {\mathrm e}^{a \textit {\_Z}} c_{1} a x +2 c_{2}^{2} a^{2} {\mathrm e}^{2 a \textit {\_Z}}+4 c_{2} a^{2} x \,{\mathrm e}^{2 a \textit {\_Z}}+2 a^{2} x^{2} {\mathrm e}^{2 a \textit {\_Z}}-\sin \left (\textit {\_Z} \right )^{2} c_{1}^{2}+c_{2}^{2} {\mathrm e}^{2 a \textit {\_Z}}+2 c_{2} x \,{\mathrm e}^{2 a \textit {\_Z}}+x^{2} {\mathrm e}^{2 a \textit {\_Z}}\right )\right )d x +c_3 \\
\end{align*}
✓ Mathematica. Time used: 0.936 (sec). Leaf size: 191
ode=(-a - 3*D[y[x],x])*D[y[x],{x,2}]^2 + (1 + D[y[x],x]^2)*Derivative[3][y][x] == 0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \int _1^x\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\exp \left (-\int _1^{K[2]}\frac {a+3 K[1]}{K[1]^2+1}dK[1]\right )}{c_1}dK[2]\&\right ][c_2+K[3]]dK[3]+c_3 \\
y(x)\to \text {Indeterminate} \\
y(x)\to \int _1^x\text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\exp \left (-\int _1^{K[2]}\frac {a+3 K[1]}{K[1]^2+1}dK[1]\right )}{c_1}dK[2]\&\right ][c_2+K[3]]dK[3]+c_3 \\
y(x)\to \int _1^x\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\exp \left (-\int _1^{K[2]}\frac {a+3 K[1]}{K[1]^2+1}dK[1]\right )}{c_1}dK[2]\&\right ][c_2+K[3]]dK[3]+c_3 \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq((-a - 3*Derivative(y(x), x))*Derivative(y(x), (x, 2))**2 + (Derivative(y(x), x)**2 + 1)*Derivative(y(x), (x, 3)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -(sqrt(4*a*Derivative(y(x), (x, 2))**2*Derivative(y(x), (x, 3)) + 9*Derivative(y(x), (x, 2))**4 - 4*Derivative(y(x), (x, 3))**2) + 3*Derivative(y(x), (x, 2))**2)/(2*Derivative(y(x), (x, 3))) + Derivative(y(x), x) cannot be solved by the factorable group method