30.6.28 problem 29

Internal problem ID [7563]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 2, First order differential equations. Review problems. page 79
Problem number : 29
Date solved : Tuesday, September 30, 2025 at 04:51:49 PM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} 4 x y^{3}-9 y^{2}+4 x y^{2}+\left (3 x^{2} y^{2}-6 x y+2 x^{2} y\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 568

ode:=4*x*y(x)^3-9*y(x)^2+4*x*y(x)^2+(3*x^2*y(x)^2-6*x*y(x)+2*x^2*y(x))*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= 0 \\ y &= \frac {\frac {{\left (\left (-x^{4}+9 x^{3}+8 \sqrt {5}\, \sqrt {c_1 \left (-x^{4}+9 x^{3}-27 x^{2}+80 c_1 +27 x \right )}-27 x^{2}+160 c_1 +27 x \right ) x^{2}\right )}^{{1}/{3}}}{3 x}+\frac {\left (x -3\right )^{2} x}{3 {\left (\left (-x^{4}+9 x^{3}+8 \sqrt {5}\, \sqrt {c_1 \left (-x^{4}+9 x^{3}-27 x^{2}+80 c_1 +27 x \right )}-27 x^{2}+160 c_1 +27 x \right ) x^{2}\right )}^{{1}/{3}}}-\frac {x}{3}+1}{x} \\ y &= \frac {\left (-i \sqrt {3}-1\right ) {\left (-\left (x^{4}-9 x^{3}+27 x^{2}-8 \sqrt {5}\, \sqrt {-c_1 \left (x^{4}-9 x^{3}+27 x^{2}-80 c_1 -27 x \right )}-27 x -160 c_1 \right ) x^{2}\right )}^{{2}/{3}}+\left (-2 {\left (-\left (x^{4}-9 x^{3}+27 x^{2}-8 \sqrt {5}\, \sqrt {-c_1 \left (x^{4}-9 x^{3}+27 x^{2}-80 c_1 -27 x \right )}-27 x -160 c_1 \right ) x^{2}\right )}^{{1}/{3}}+\left (i \sqrt {3}-1\right ) x \left (x -3\right )\right ) x \left (x -3\right )}{6 {\left (-\left (x^{4}-9 x^{3}+27 x^{2}-8 \sqrt {5}\, \sqrt {-c_1 \left (x^{4}-9 x^{3}+27 x^{2}-80 c_1 -27 x \right )}-27 x -160 c_1 \right ) x^{2}\right )}^{{1}/{3}} x^{2}} \\ y &= -\frac {\left (1-i \sqrt {3}\right ) {\left (-\left (x^{4}-9 x^{3}+27 x^{2}-8 \sqrt {5}\, \sqrt {-c_1 \left (x^{4}-9 x^{3}+27 x^{2}-80 c_1 -27 x \right )}-27 x -160 c_1 \right ) x^{2}\right )}^{{2}/{3}}+x \left (2 {\left (-\left (x^{4}-9 x^{3}+27 x^{2}-8 \sqrt {5}\, \sqrt {-c_1 \left (x^{4}-9 x^{3}+27 x^{2}-80 c_1 -27 x \right )}-27 x -160 c_1 \right ) x^{2}\right )}^{{1}/{3}}+\left (1+i \sqrt {3}\right ) x \left (x -3\right )\right ) \left (x -3\right )}{6 {\left (-\left (x^{4}-9 x^{3}+27 x^{2}-8 \sqrt {5}\, \sqrt {-c_1 \left (x^{4}-9 x^{3}+27 x^{2}-80 c_1 -27 x \right )}-27 x -160 c_1 \right ) x^{2}\right )}^{{1}/{3}} x^{2}} \\ \end{align*}

✓ Mathematica. Time used: 54.895 (sec). Leaf size: 856

ode=(4*x*y[x]^3 -9*y[x]^2 +4*x*y[x]^2)+( 3*x^2*y[x]^2 - 6*x*y[x]+2*x^2*y[x])*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to 0\\ y(x)&\to \frac {1}{6} \left (2^{2/3} e^{-\frac {c_1}{3}} \sqrt [3]{\frac {-2 e^{c_1} x (x-3)^3+3 \sqrt {3} \sqrt {e^{\frac {4 c_1}{3}} \left (27+4 e^{\frac {c_1}{3}} x (x-3)^3\right )}-27 e^{\frac {2 c_1}{3}}}{x^4}}+\frac {2 \sqrt [3]{2} e^{\frac {c_1}{3}} (x-3)^2}{x^2 \sqrt [3]{\frac {-2 e^{c_1} x (x-3)^3+3 \sqrt {3} \sqrt {e^{\frac {4 c_1}{3}} \left (27+4 e^{\frac {c_1}{3}} x (x-3)^3\right )}-27 e^{\frac {2 c_1}{3}}}{x^4}}}+\frac {6}{x}-2\right )\\ y(x)&\to \frac {1}{12} \left (i 2^{2/3} \left (\sqrt {3}+i\right ) e^{-\frac {c_1}{3}} \sqrt [3]{\frac {-2 e^{c_1} x (x-3)^3+3 \sqrt {3} \sqrt {e^{\frac {4 c_1}{3}} \left (27+4 e^{\frac {c_1}{3}} x (x-3)^3\right )}-27 e^{\frac {2 c_1}{3}}}{x^4}}-\frac {2 i \sqrt [3]{2} \left (\sqrt {3}-i\right ) e^{\frac {c_1}{3}} (x-3)^2}{x^2 \sqrt [3]{\frac {-2 e^{c_1} x (x-3)^3+3 \sqrt {3} \sqrt {e^{\frac {4 c_1}{3}} \left (27+4 e^{\frac {c_1}{3}} x (x-3)^3\right )}-27 e^{\frac {2 c_1}{3}}}{x^4}}}+\frac {12}{x}-4\right )\\ y(x)&\to \frac {1}{12} \left (-i 2^{2/3} \left (\sqrt {3}-i\right ) e^{-\frac {c_1}{3}} \sqrt [3]{\frac {-2 e^{c_1} x (x-3)^3+3 \sqrt {3} \sqrt {e^{\frac {4 c_1}{3}} \left (27+4 e^{\frac {c_1}{3}} x (x-3)^3\right )}-27 e^{\frac {2 c_1}{3}}}{x^4}}+\frac {2 i \sqrt [3]{2} \left (\sqrt {3}+i\right ) e^{\frac {c_1}{3}} (x-3)^2}{x^2 \sqrt [3]{\frac {-2 e^{c_1} x (x-3)^3+3 \sqrt {3} \sqrt {e^{\frac {4 c_1}{3}} \left (27+4 e^{\frac {c_1}{3}} x (x-3)^3\right )}-27 e^{\frac {2 c_1}{3}}}{x^4}}}+\frac {12}{x}-4\right )\\ y(x)&\to 0\\ y(x)&\to \frac {\left (6+\left (-i \sqrt {3} \sqrt [3]{-\frac {(x-3)^3}{x^3}}+\sqrt [3]{-\frac {(x-3)^3}{x^3}}-2\right ) x\right ) \left (\sqrt [3]{-\frac {(x-3)^3}{x^3}} x+x-3\right )}{6 (x-3) x}\\ y(x)&\to \frac {\left (6+\left (i \sqrt {3} \sqrt [3]{-\frac {(x-3)^3}{x^3}}+\sqrt [3]{-\frac {(x-3)^3}{x^3}}-2\right ) x\right ) \left (\sqrt [3]{-\frac {(x-3)^3}{x^3}} x+x-3\right )}{6 (x-3) x}\\ y(x)&\to -\frac {3 \left (\sqrt [3]{-\frac {(x-3)^3}{x^3}}-2\right ) x+\left (\left (-\frac {(x-3)^3}{x^3}\right )^{2/3}-\sqrt [3]{-\frac {(x-3)^3}{x^3}}+1\right ) x^2+9}{3 (x-3) x} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x*y(x)**3 + 4*x*y(x)**2 + (3*x**2*y(x)**2 + 2*x**2*y(x) - 6*x*y(x))*Derivative(y(x), x) - 9*y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out