50.3.27 problem 27

Internal problem ID [10171]
Book : Own collection of miscellaneous problems
Section : section 3.0
Problem number : 27
Date solved : Tuesday, September 30, 2025 at 07:10:58 PM
CAS classification : [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]

\begin{align*} \left (x^{2}+1\right ) y^{\prime \prime }+{y^{\prime }}^{3}&=0 \end{align*}

✓ Maple. Time used: 0.022 (sec). Leaf size: 33

ode:=(x^2+1)*diff(diff(y(x),x),x)+diff(y(x),x)^3 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \int \frac {1}{\sqrt {c_1 +2 \arctan \left (x \right )}}d x +c_2 \\ y &= -\int \frac {1}{\sqrt {c_1 +2 \arctan \left (x \right )}}d x +c_2 \\ \end{align*}

✓ Mathematica. Time used: 10.297 (sec). Leaf size: 100

ode=(1+x^2)*D[y[x],{x,2}]+D[y[x],x]^3==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \int _1^x-\frac {1}{\sqrt {2} \sqrt {-c_1-\int _1^{K[2]}-\frac {1}{K[1]^2+1}dK[1]}}dK[2]+c_2\\ y(x)&\to \int _1^x\frac {1}{\sqrt {2} \sqrt {-c_1-\int _1^{K[3]}-\frac {1}{K[1]^2+1}dK[1]}}dK[3]+c_2 \end{align*}

✓ Sympy. Time used: 82.406 (sec). Leaf size: 143

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((x**2 + 1)*Derivative(y(x), (x, 2)) + Derivative(y(x), x)**3,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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