56.3.27 problem 27
Internal
problem
ID
[8885]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
3.0
Problem
number
:
27
Date
solved
:
Wednesday, March 05, 2025 at 07:07:24 AM
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: 0.013 (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: 62.291 (sec). Leaf size: 59
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 \arctan (K[1])-2 c_1}}dK[1]+c_2 \\
y(x)\to \int _1^x\frac {1}{\sqrt {2 \arctan (K[2])-2 c_1}}dK[2]+c_2 \\
\end{align*}
✓ Sympy. Time used: 90.525 (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)
\[
\left [ y{\left (x \right )} = C_{1} - \frac {\sqrt {2} \int \sqrt {- \frac {1}{C_{2} - \operatorname {atan}{\left (x \right )}}}\, dx}{2}, \ y{\left (x \right )} = C_{1} + \frac {\sqrt {2} \int \sqrt {- \frac {1}{C_{2} - \operatorname {atan}{\left (x \right )}}}\, dx}{2}, \ y{\left (x \right )} = C_{1} - \frac {\sqrt {2} \int \sqrt {- \frac {1}{C_{2} - \operatorname {atan}{\left (x \right )}}}\, dx}{2}, \ y{\left (x \right )} = C_{1} + \frac {\sqrt {2} \int \sqrt {- \frac {1}{C_{2} - \operatorname {atan}{\left (x \right )}}}\, dx}{2}, \ y{\left (x \right )} = C_{1} - \frac {\sqrt {2} \int \sqrt {- \frac {1}{C_{2} - \operatorname {atan}{\left (x \right )}}}\, dx}{2}, \ y{\left (x \right )} = C_{1} + \frac {\sqrt {2} \int \sqrt {- \frac {1}{C_{2} - \operatorname {atan}{\left (x \right )}}}\, dx}{2}\right ]
\]