9.1.28 problem 28

Internal problem ID [2868]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 5, page 21
Problem number : 28
Date solved : Tuesday, September 30, 2025 at 05:55:36 AM
CAS classification : [_separable]

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

With initial conditions

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

✓ Maple. Time used: 0.239 (sec). Leaf size: 45

ode:=(x^2+3*x)*diff(y(x),x) = y(x)^3+2*y(x); 
ic:=[y(1) = 1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {4 \sqrt {3 \,2^{{1}/{3}} \left (x +3\right )^{{4}/{3}} x^{{4}/{3}}-8 x^{{8}/{3}}}}{3 \,2^{{1}/{3}} \left (x +3\right )^{{4}/{3}}-8 x^{{4}/{3}}} \]

✓ Mathematica. Time used: 34.819 (sec). Leaf size: 266

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

\begin{align*} y(x)&\to \frac {2\ 2^{5/6} \sqrt {-x^{4/3} \left (32 \sqrt [3]{2} x^{8/3}+12\ 2^{2/3} \sqrt [3]{x+3} x^{7/3}+36\ 2^{2/3} \sqrt [3]{x+3} x^{4/3}+9 (x+3)^{2/3} x^2+54 (x+3)^{2/3} x+81 (x+3)^{2/3}\right )}}{\sqrt {229 x^4-324 x^3-1458 x^2-2916 x-2187}}\\ y(x)&\to \frac {2 \sqrt {-x^{4/3} \left (64 x^{8/3}+24 \sqrt [3]{2} \sqrt [3]{x+3} x^{7/3}+72 \sqrt [3]{2} \sqrt [3]{x+3} x^{4/3}+9\ 2^{2/3} (x+3)^{2/3} x^2+54\ 2^{2/3} (x+3)^{2/3} x+81\ 2^{2/3} (x+3)^{2/3}\right )}}{\sqrt {\frac {229 x^4}{2}-162 x^3-729 x^2-1458 x-\frac {2187}{2}}} \end{align*}

✗ Sympy

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

Timed Out