56.3.23 problem 23

Internal problem ID [8881]
Book : Own collection of miscellaneous problems
Section : section 3.0
Problem number : 23
Date solved : Sunday, March 30, 2025 at 01:52:12 PM
CAS classification : [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]

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

✓ Maple. Time used: 0.082 (sec). Leaf size: 22

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

\[ y = 2 \int \frac {1}{\ln \left (x^{2}+1\right )+2 c_1}d x +c_2 \]

✓ Mathematica. Time used: 60.266 (sec). Leaf size: 33

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

\[ y(x)\to \int _1^x-\frac {2}{2 c_1-\log \left (K[1]^2+1\right )}dK[1]+c_2 \]

✓ Sympy. Time used: 2.903 (sec). Leaf size: 34

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

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