89.2.27 problem 29

Internal problem ID [24292]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 2. Equations of the first order and first degree. Exercises at page 27
Problem number : 29
Date solved : Thursday, October 02, 2025 at 10:07:48 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

With initial conditions

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

✓ Maple. Time used: 8.723 (sec). Leaf size: 359

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

\begin{align*} y &= \frac {2^{{5}/{6}} \sqrt {\left (-2 x^{6}+16 \sqrt {-x^{6}+16}+64\right )^{{1}/{3}} \left (2^{{2}/{3}} x^{4}-2^{{1}/{3}} x^{2} \left (-2 x^{6}+16 \sqrt {-x^{6}+16}+64\right )^{{1}/{3}}+\left (-2 x^{6}+16 \sqrt {-x^{6}+16}+64\right )^{{2}/{3}}\right )}}{4 \left (-2 x^{6}+16 \sqrt {-x^{6}+16}+64\right )^{{1}/{3}}} \\ y &= \frac {2^{{5}/{6}} \sqrt {-\left (-2 x^{6}+16 \sqrt {-x^{6}+16}+64\right )^{{1}/{3}} \left (2^{{2}/{3}} x^{4}-2^{{1}/{3}} x^{2} \left (-2 x^{6}+16 \sqrt {-x^{6}+16}+64\right )^{{1}/{3}}+\left (-2 x^{6}+16 \sqrt {-x^{6}+16}+64\right )^{{2}/{3}}\right ) \left (1+i \sqrt {3}\right )}}{4 \sqrt {-1-i \sqrt {3}}\, \left (-2 x^{6}+16 \sqrt {-x^{6}+16}+64\right )^{{1}/{3}}} \\ y &= \frac {2^{{5}/{6}} \sqrt {\left (i \sqrt {3}-1\right ) \left (2^{{2}/{3}} x^{4}-2^{{1}/{3}} x^{2} \left (-2 x^{6}+16 \sqrt {-x^{6}+16}+64\right )^{{1}/{3}}+\left (-2 x^{6}+16 \sqrt {-x^{6}+16}+64\right )^{{2}/{3}}\right ) \left (-2 x^{6}+16 \sqrt {-x^{6}+16}+64\right )^{{1}/{3}}}}{4 \sqrt {i \sqrt {3}-1}\, \left (-2 x^{6}+16 \sqrt {-x^{6}+16}+64\right )^{{1}/{3}}} \\ \end{align*}

✓ Mathematica. Time used: 58.541 (sec). Leaf size: 73

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

\begin{align*} y(x)&\to \frac {1}{2} \sqrt {\sqrt [3]{8 \left (\sqrt {16-x^6}+4\right )-x^6}-x^2+\frac {x^4}{\sqrt [3]{8 \left (\sqrt {16-x^6}+4\right )-x^6}}} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*y(x) + (2*x**2 + 4*y(x)**2)*Derivative(y(x), x),0) 
ics = {y(0): 1} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out