87.7.11 problem 14

Internal problem ID [23352]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 1. Elementary methods. First order differential equations. Exercise at page 57
Problem number : 14
Date solved : Thursday, October 02, 2025 at 09:39:27 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} y^{\prime \prime }&=\frac {1+{y^{\prime }}^{2}}{y} \end{align*}

✓ Maple. Time used: 0.013 (sec). Leaf size: 27

ode:=diff(diff(y(x),x),x) = (1+diff(y(x),x)^2)/y(x); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= c_1 \cosh \left (\frac {c_2 +x}{c_1}\right ) \\ y &= c_1 \cosh \left (\frac {c_2 +x}{c_1}\right ) \\ \end{align*}

✓ Mathematica. Time used: 60.125 (sec). Leaf size: 128

ode=D[y[x],{x,2}]==(1+D[y[x],x]^2)/y[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {e^{-c_1} \tanh \left (\sqrt {e^{2 c_1}} (x+c_2)\right )}{\sqrt {-\text {sech}^2\left (\sqrt {e^{2 c_1}} (x+c_2)\right )}}\\ y(x)&\to \text {InverseFunction}\left [-\frac {\sqrt {1-\text {$\#$1}^2 e^{2 c_1}} \text {arcsinh}\left (\text {$\#$1} \sqrt {-e^{2 c_1}}\right )}{\sqrt {-e^{2 c_1}} \sqrt {-1+\text {$\#$1}^2 e^{2 c_1}}}\&\right ][x+c_2] \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-(Derivative(y(x), x)**2 + 1)/y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -sqrt(y(x)*Derivative(y(x), (x, 2)) - 1) + Derivative(y(x), x) c