15.18.34 problem 34

Internal problem ID [3277]
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 : 34
Date solved : Sunday, March 30, 2025 at 01:26:16 AM
CAS classification : [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]

\begin{align*} \left (x^{2}+1\right ) y^{\prime \prime }+1+{y^{\prime }}^{2}&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=1 \end{align*}

✓ Maple. Time used: 0.163 (sec). Leaf size: 20

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

\[ y = -x +2 \ln \left (-x -1\right )+1-2 i \pi \]

✓ Mathematica. Time used: 7.253 (sec). Leaf size: 23

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

\[ y(x)\to -x+2 \log (-x-1)-2 i \pi +1 \]

✓ Sympy. Time used: 1.562 (sec). Leaf size: 48

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

\[ \left [ y{\left (x \right )} = \int \cot {\left (\operatorname {atan}{\left (x \right )} + \frac {\pi }{4} \right )}\, dx - \int \limits ^{0} \cot {\left (\operatorname {atan}{\left (x \right )} + \frac {\pi }{4} \right )}\, dx + 1, \ y{\left (x \right )} = \int \cot {\left (\operatorname {atan}{\left (x \right )} + \frac {\pi }{4} \right )}\, dx - \int \limits ^{0} \cot {\left (\operatorname {atan}{\left (x \right )} + \frac {\pi }{4} \right )}\, dx + 1\right ] \]