62.38.1 problem Ex 1

Internal problem ID [12941]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter IX, Miscellaneous methods for solving equations of higher order than first. Article 62. Summary. Page 144
Problem number : Ex 1
Date solved : Monday, March 31, 2025 at 07:27:35 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.018 (sec). Leaf size: 17

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

\[ y = -\ln \left (c_1 \sin \left (x \right )-c_2 \cos \left (x \right )\right ) \]

✓ Mathematica. Time used: 0.8 (sec). Leaf size: 38

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

\[ y(x)\to \int _1^x\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1]^2+1}dK[1]\&\right ][c_1+K[2]]dK[2]+c_2 \]

✓ Sympy. Time used: 1.193 (sec). Leaf size: 31

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

\[ \left [ y{\left (x \right )} = C_{1} + \frac {\log {\left (\tan ^{2}{\left (C_{2} - x \right )} + 1 \right )}}{2}, \ y{\left (x \right )} = C_{1} + \frac {\log {\left (\tan ^{2}{\left (C_{2} - x \right )} + 1 \right )}}{2}\right ] \]