83.5.4 problem 1 (d)

Internal problem ID [21939]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Chapter III. First order differential equations of the first degree. Ex. VI at page 47
Problem number : 1 (d)
Date solved : Thursday, October 02, 2025 at 08:17:59 PM
CAS classification : [[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} 3 y+2 x y^{\prime }+4 x y^{2}+3 x^{2} y y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 375

ode:=3*y(x)+2*x*diff(y(x),x)+4*x*y(x)^2+3*x^2*y(x)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {2 x^{2} 2^{{1}/{3}}-2 x {\left (\left (3 \sqrt {3}\, \sqrt {27 c_1^{2}-4 c_1 x}+27 c_1 -2 x \right ) x^{2}\right )}^{{1}/{3}}+2^{{2}/{3}} {\left (\left (3 \sqrt {3}\, \sqrt {27 c_1^{2}-4 c_1 x}+27 c_1 -2 x \right ) x^{2}\right )}^{{2}/{3}}}{6 x^{2} {\left (\left (3 \sqrt {3}\, \sqrt {27 c_1^{2}-4 c_1 x}+27 c_1 -2 x \right ) x^{2}\right )}^{{1}/{3}}} \\ y &= \frac {-\frac {2^{{2}/{3}} \left (1+i \sqrt {3}\right ) {\left (\left (3 \sqrt {3}\, \sqrt {27 c_1^{2}-4 c_1 x}+27 c_1 -2 x \right ) x^{2}\right )}^{{2}/{3}}}{2}+x \left (-2 {\left (\left (3 \sqrt {3}\, \sqrt {27 c_1^{2}-4 c_1 x}+27 c_1 -2 x \right ) x^{2}\right )}^{{1}/{3}}+x 2^{{1}/{3}} \left (i \sqrt {3}-1\right )\right )}{6 {\left (\left (3 \sqrt {3}\, \sqrt {27 c_1^{2}-4 c_1 x}+27 c_1 -2 x \right ) x^{2}\right )}^{{1}/{3}} x^{2}} \\ y &= -\frac {-\frac {2^{{2}/{3}} \left (i \sqrt {3}-1\right ) {\left (\left (3 \sqrt {3}\, \sqrt {27 c_1^{2}-4 c_1 x}+27 c_1 -2 x \right ) x^{2}\right )}^{{2}/{3}}}{2}+x \left (2 {\left (\left (3 \sqrt {3}\, \sqrt {27 c_1^{2}-4 c_1 x}+27 c_1 -2 x \right ) x^{2}\right )}^{{1}/{3}}+x 2^{{1}/{3}} \left (1+i \sqrt {3}\right )\right )}{6 {\left (\left (3 \sqrt {3}\, \sqrt {27 c_1^{2}-4 c_1 x}+27 c_1 -2 x \right ) x^{2}\right )}^{{1}/{3}} x^{2}} \\ \end{align*}

✓ Mathematica. Time used: 65.83 (sec). Leaf size: 401

ode=3*y[x]+2*x*D[y[x],x]+4*x*y[x]^2+3*x^2*y[x]*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{6} \left (2^{2/3} \sqrt [3]{\frac {-2 x+3 \sqrt {3} \sqrt {e^{c_1} \left (-4 x+27 e^{c_1}\right )}+27 e^{c_1}}{x^4}}+\frac {2 \sqrt [3]{2}}{x^2 \sqrt [3]{\frac {-2 x+3 \sqrt {3} \sqrt {e^{c_1} \left (-4 x+27 e^{c_1}\right )}+27 e^{c_1}}{x^4}}}-\frac {2}{x}\right )\\ y(x)&\to \frac {1}{12} \left (i 2^{2/3} \left (\sqrt {3}+i\right ) \sqrt [3]{\frac {-2 x+3 \sqrt {3} \sqrt {e^{c_1} \left (-4 x+27 e^{c_1}\right )}+27 e^{c_1}}{x^4}}-\frac {2 \sqrt [3]{2} \left (1+i \sqrt {3}\right )}{x^2 \sqrt [3]{\frac {-2 x+3 \sqrt {3} \sqrt {e^{c_1} \left (-4 x+27 e^{c_1}\right )}+27 e^{c_1}}{x^4}}}-\frac {4}{x}\right )\\ y(x)&\to \frac {1}{12} \left (-2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{\frac {-2 x+3 \sqrt {3} \sqrt {e^{c_1} \left (-4 x+27 e^{c_1}\right )}+27 e^{c_1}}{x^4}}+\frac {2 i \sqrt [3]{2} \left (\sqrt {3}+i\right )}{x^2 \sqrt [3]{\frac {-2 x+3 \sqrt {3} \sqrt {e^{c_1} \left (-4 x+27 e^{c_1}\right )}+27 e^{c_1}}{x^4}}}-\frac {4}{x}\right ) \end{align*}

✓ Sympy. Time used: 31.153 (sec). Leaf size: 410

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(3*x**2*y(x)*Derivative(y(x), x) + 4*x*y(x)**2 + 2*x*Derivative(y(x), x) + 3*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = - \frac {2^{\frac {2}{3}} \sqrt [3]{- \frac {27 C_{1}}{x^{4}} + 3 \sqrt {3} \sqrt {\frac {C_{1} \left (\frac {27 C_{1}}{x} - 4\right )}{x^{7}}} + \frac {2}{x^{3}}}}{6} - \frac {1}{3 x} - \frac {\sqrt [3]{2}}{3 x^{2} \sqrt [3]{- \frac {27 C_{1}}{x^{4}} + 3 \sqrt {3} \sqrt {\frac {C_{1} \left (\frac {27 C_{1}}{x} - 4\right )}{x^{7}}} + \frac {2}{x^{3}}}}, \ y{\left (x \right )} = \frac {\sqrt [3]{- \frac {27 C_{1}}{2 x^{4}} + \frac {3 \sqrt {3} \sqrt {\frac {C_{1} \left (\frac {27 C_{1}}{x} - 4\right )}{x^{7}}}}{2} + \frac {1}{x^{3}}}}{6} - \frac {\sqrt {3} i \sqrt [3]{- \frac {27 C_{1}}{2 x^{4}} + \frac {3 \sqrt {3} \sqrt {\frac {C_{1} \left (\frac {27 C_{1}}{x} - 4\right )}{x^{7}}}}{2} + \frac {1}{x^{3}}}}{6} - \frac {1}{3 x} + \frac {2}{3 x^{2} \left (1 - \sqrt {3} i\right ) \sqrt [3]{- \frac {27 C_{1}}{2 x^{4}} + \frac {3 \sqrt {3} \sqrt {\frac {C_{1} \left (\frac {27 C_{1}}{x} - 4\right )}{x^{7}}}}{2} + \frac {1}{x^{3}}}}, \ y{\left (x \right )} = \frac {\sqrt [3]{- \frac {27 C_{1}}{2 x^{4}} + \frac {3 \sqrt {3} \sqrt {\frac {C_{1} \left (\frac {27 C_{1}}{x} - 4\right )}{x^{7}}}}{2} + \frac {1}{x^{3}}}}{6} + \frac {\sqrt {3} i \sqrt [3]{- \frac {27 C_{1}}{2 x^{4}} + \frac {3 \sqrt {3} \sqrt {\frac {C_{1} \left (\frac {27 C_{1}}{x} - 4\right )}{x^{7}}}}{2} + \frac {1}{x^{3}}}}{6} - \frac {1}{3 x} + \frac {2}{3 x^{2} \left (1 + \sqrt {3} i\right ) \sqrt [3]{- \frac {27 C_{1}}{2 x^{4}} + \frac {3 \sqrt {3} \sqrt {\frac {C_{1} \left (\frac {27 C_{1}}{x} - 4\right )}{x^{7}}}}{2} + \frac {1}{x^{3}}}}\right ] \]