87.7.13 problem 16

Internal problem ID [23354]
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 : 16
Date solved : Thursday, October 02, 2025 at 09:39:32 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.007 (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: 1.441 (sec). Leaf size: 16

ode=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 c_2-\log (\cos (x+c_1)) \end{align*}

✓ Sympy. Time used: 0.901 (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)
 

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