54.8.13 problem 1849

Internal problem ID [13073]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 7, non-linear third and higher order
Problem number : 1849
Date solved : Wednesday, October 01, 2025 at 03:34:08 AM
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 {b^{2} {y^{\prime \prime }}^{2}+1}&=0 \end{align*}

✓ Maple. Time used: 0.097 (sec). Leaf size: 293

ode:=diff(diff(y(x),x),x)*diff(diff(diff(y(x),x),x),x)-a*(b^2*diff(diff(y(x),x),x)^2+1)^(1/2) = 0; 
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: 60.352 (sec). Leaf size: 194

ode=-(a*Sqrt[1 + b^2*D[y[x],{x,2}]^2]) + 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\frac {\log \left (b^2 \left ((c_1+a K[1]) b^2+\sqrt {b^4 (c_1+a K[1]){}^2-1}\right )\right )-b^2 (c_1+a K[1]) \sqrt {b^4 (c_1+a K[1]){}^2-1}}{2 a b^3}dK[1]+c_3 x+c_2\\ y(x)&\to \int _1^x\frac {b^2 (c_1+a K[2]) \sqrt {b^4 (c_1+a K[2]){}^2-1}-\log \left (b^2 \left ((c_1+a K[2]) b^2+\sqrt {b^4 (c_1+a K[2]){}^2-1}\right )\right )}{2 a b^3}dK[2]+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