15.18.22 problem 22

Internal problem ID [3265]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 35, page 157
Problem number : 22
Date solved : Sunday, March 30, 2025 at 01:25:40 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

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

✓ Maple. Time used: 0.039 (sec). Leaf size: 28

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

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

✓ Mathematica. Time used: 13.494 (sec). Leaf size: 84

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

\begin{align*} y(x)\to -\frac {\sinh \left (\sqrt {e^{2 c_1}} (x+c_2)\right )}{\sqrt {e^{2 c_1}}} \\ y(x)\to \frac {\sinh \left (\sqrt {e^{2 c_1}} (x+c_2)\right )}{\sqrt {e^{2 c_1}}} \\ y(x)\to -x-c_2 \\ y(x)\to x+c_2 \\ \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x)*Derivative(y(x), (x, 2)) - Derivative(y(x), x)**2 + 1,0) 
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) cannot be solved by the factorable group method