9.8.42 problem 44

Internal problem ID [3045]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 12, page 46
Problem number : 44
Date solved : Tuesday, September 30, 2025 at 06:24:37 AM
CAS classification : [[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} \left (-2 x^{2}-3 x y\right ) y^{\prime }+y^{2}&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.451 (sec). Leaf size: 100

ode:=(-2*x^2-3*x*y(x))*diff(y(x),x)+y(x)^2 = 0; 
ic:=[y(1) = 1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {\left (-x^{3}+3 \sqrt {-2 x^{4}+27 x^{2}}\, \sqrt {3}+27 x \right )^{{2}/{3}}-x \left (-x^{3}+3 \sqrt {-2 x^{4}+27 x^{2}}\, \sqrt {3}+27 x \right )^{{1}/{3}}+x^{2}}{3 \left (-x^{3}+3 \sqrt {-2 x^{4}+27 x^{2}}\, \sqrt {3}+27 x \right )^{{1}/{3}}} \]

✓ Mathematica. Time used: 60.201 (sec). Leaf size: 77

ode=(-2*x^2-3*x*y[x])*D[y[x],x]+y[x]^2==0; 
ic={y[1]==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{3} \left (\frac {x^2}{\sqrt [3]{-x^3+3 \sqrt {81 x^2-6 x^4}+27 x}}+\sqrt [3]{-x^3+3 \sqrt {81 x^2-6 x^4}+27 x}-x\right ) \end{align*}

✓ Sympy. Time used: 28.990 (sec). Leaf size: 150

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

\[ y{\left (x \right )} = \frac {\frac {4 \sqrt [3]{2} x^{2}}{\sqrt [3]{2 x^{3} - 54 x + 3 \sqrt {6} \sqrt {x^{2} \left (54 - 4 x^{2}\right )}}} - 2 x + 2 \sqrt {3} i x - 2^{\frac {2}{3}} \sqrt [3]{2 x^{3} - 54 x + 3 \sqrt {6} \sqrt {x^{2} \left (54 - 4 x^{2}\right )}} - 2^{\frac {2}{3}} \sqrt {3} i \sqrt [3]{2 x^{3} - 54 x + 3 \sqrt {6} \sqrt {x^{2} \left (54 - 4 x^{2}\right )}}}{6 \left (1 - \sqrt {3} i\right )} \]