56.5.15 problem 15

Internal problem ID [8976]
Book : Own collection of miscellaneous problems
Section : section 5.0
Problem number : 15
Date solved : Sunday, March 30, 2025 at 01:57:41 PM
CAS classification : [_quadrature]

\begin{align*} h^{2}+\frac {2 a h}{\sqrt {1+{h^{\prime }}^{2}}}&=b^{2} \end{align*}

✓ Maple. Time used: 1.033 (sec). Leaf size: 103

ode:=h(u)^2+2*a*h(u)/(1+diff(h(u),u)^2)^(1/2) = b^2; 
dsolve(ode,h(u), singsol=all);
 

\begin{align*} u -\int _{}^{h}\frac {\textit {\_a}^{2}-b^{2}}{\sqrt {-\textit {\_a}^{4}+\left (4 a^{2}+2 b^{2}\right ) \textit {\_a}^{2}-b^{4}}}d \textit {\_a} -c_1 &= 0 \\ u +\int _{}^{h}\frac {\textit {\_a}^{2}-b^{2}}{\sqrt {-\textit {\_a}^{4}+\left (4 a^{2}+2 b^{2}\right ) \textit {\_a}^{2}-b^{4}}}d \textit {\_a} -c_1 &= 0 \\ \end{align*}

✓ Mathematica. Time used: 35.557 (sec). Leaf size: 913

ode=h[u]^2 + 2*a*h[u]/Sqrt[1 + (D[ h[u],u])^2] == b^2; 
ic={}; 
DSolve[{ode,ic},h[u],u,IncludeSingularSolutions->True]
 

\begin{align*} \text {Solution too large to show}\end{align*}

✗ Sympy

from sympy import * 
u = symbols("u") 
a = symbols("a") 
b = symbols("b") 
h = Function("h") 
ode = Eq(2*a*h(u)/sqrt(Derivative(h(u), u)**2 + 1) - b**2 + h(u)**2,0) 
ics = {} 
dsolve(ode,func=h(u),ics=ics)
 

Timed Out