49.23.7 problem 2

Internal problem ID [7765]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 6. Existence and uniqueness of solutions to systems and nth order equations. Page 238
Problem number : 2
Date solved : Wednesday, March 05, 2025 at 04:54:54 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]

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

With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Maple. Time used: 0.039 (sec). Leaf size: 7

ode:=diff(diff(y(x),x),x) = 1+diff(y(x),x)^2; 
ic:=y(0) = 0, D(y)(0) = 0; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = \ln \left (\sec \left (x \right )\right ) \]

✓ Mathematica. Time used: 2.41 (sec). Leaf size: 27

ode=D[y[x],{x,2}]==1+(D[y[x],x])^2; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\log (-\cos (x))+i \pi \\ y(x)\to -\log (\cos (x)) \\ \end{align*}

✓ Sympy. Time used: 1.219 (sec). Leaf size: 24

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

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