56.3.27 problem 27

Internal problem ID [8885]
Book : Own collection of miscellaneous problems
Section : section 3.0
Problem number : 27
Date solved : Monday, January 27, 2025 at 05:18:29 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*}

✓ Solution by Maple

Time used: 0.013 (sec). Leaf size: 33

dsolve((1+x^2)*diff(y(x),x$2)+diff(y(x),x)^3=0,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*}

✓ Solution by Mathematica

Time used: 62.291 (sec). Leaf size: 59

DSolve[(1+x^2)*D[y[x],{x,2}]+D[y[x],x]^3==0,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*}