80.3.26 problem 27

Internal problem ID [21190]
Book : A Textbook on Ordinary Differential Equations by Shair Ahmad and Antonio Ambrosetti. Second edition. ISBN 978-3-319-16407-6. Springer 2015
Section : Chapter 3. First order nonlinear differential equations. Excercise 3.7 at page 67
Problem number : 27
Date solved : Thursday, October 02, 2025 at 07:16:21 PM
CAS classification : [[_homogeneous, `class A`], _exact, _rational, _dAlembert]

\begin{align*} x^{2}+2 y x +2 y^{2}+\left (x^{2}+4 y x +5 y^{2}\right ) y^{\prime }&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.053 (sec). Leaf size: 177

ode:=x^2+2*x*y(x)+2*y(x)^2+(x^2+4*x*y(x)+5*y(x)^2)*diff(y(x),x) = 0; 
ic:=[y(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\begin{align*} y &= \frac {\left (5 \left (44+20 \sqrt {5}\right )^{{2}/{3}} \sqrt {5}-11 \left (44+20 \sqrt {5}\right )^{{2}/{3}}-4 \left (44+20 \sqrt {5}\right )^{{1}/{3}}-16\right ) x}{40} \\ y &= -\frac {\left (i \sqrt {3}\, \left (44+20 \sqrt {5}\right )^{{2}/{3}}+4 i \sqrt {3}-\left (44+20 \sqrt {5}\right )^{{2}/{3}}+8 \left (44+20 \sqrt {5}\right )^{{1}/{3}}+4\right ) x \left (44+20 \sqrt {5}\right )^{{2}/{3}} \left (-11+5 \sqrt {5}\right )}{320} \\ y &= \frac {\left (i \sqrt {3}\, \left (44+20 \sqrt {5}\right )^{{2}/{3}}+4 i \sqrt {3}+\left (44+20 \sqrt {5}\right )^{{2}/{3}}-8 \left (44+20 \sqrt {5}\right )^{{1}/{3}}-4\right ) x \left (44+20 \sqrt {5}\right )^{{2}/{3}} \left (-11+5 \sqrt {5}\right )}{320} \\ \end{align*}

✓ Mathematica. Time used: 57.433 (sec). Leaf size: 310

ode=(x^2+2*x*y[x]+2*y[x]^2)+(x^2+4*x*y[x]+5*y[x]^2)*D[y[x],x]==0; 
ic={y[0]==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {2 \sqrt [3]{-2} x^2-4 \sqrt [3]{5 \sqrt {5} \sqrt {x^6}-11 x^3} x+(-2)^{2/3} \left (5 \sqrt {5} \sqrt {x^6}-11 x^3\right )^{2/3}}{10 \sqrt [3]{5 \sqrt {5} \sqrt {x^6}-11 x^3}}\\ y(x)&\to \frac {-2 \sqrt [3]{2} x^2-4 \sqrt [3]{5 \sqrt {5} \sqrt {x^6}-11 x^3} x+2^{2/3} \left (5 \sqrt {5} \sqrt {x^6}-11 x^3\right )^{2/3}}{10 \sqrt [3]{5 \sqrt {5} \sqrt {x^6}-11 x^3}}\\ y(x)&\to \frac {\left (5 \sqrt {5} \sqrt {x^6}-11 x^3\right )^{2/3} \text {Root}\left [\text {$\#$1}^3-32\&,2\right ]+x^2 \text {Root}\left [\text {$\#$1}^3+128\&,2\right ]-8 \sqrt [3]{5 \sqrt {5} \sqrt {x^6}-11 x^3} x}{20 \sqrt [3]{5 \sqrt {5} \sqrt {x^6}-11 x^3}} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2 + 2*x*y(x) + (x**2 + 4*x*y(x) + 5*y(x)**2)*Derivative(y(x), x) + 2*y(x)**2,0) 
ics = {y(0): 0} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out