89.9.4 problem 4

Internal problem ID [24484]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 4. Additional topics on equations of first order and first degree. Exercises at page 72
Problem number : 4
Date solved : Thursday, October 02, 2025 at 10:42:10 PM
CAS classification : [_rational, [_Abel, `2nd type`, `class A`]]

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

✓ Maple. Time used: 0.013 (sec). Leaf size: 25

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

\[ y = -\frac {x^{2}}{\operatorname {LambertW}\left (-x^{2} {\mathrm e}^{\frac {c_1}{2}+\frac {x}{2}}\right )} \]

✓ Mathematica. Time used: 4.338 (sec). Leaf size: 408

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

\[ \text {Solve}\left [-\frac {2^{2/3} \sqrt [3]{x^3 (x+4)^3} \left (x^2-y(x)\right )^3 \left (1-\frac {x (x+4) \left (x^2+2 y(x)\right )}{\sqrt [3]{x^3 (x+4)^3} \left (x^2-y(x)\right )}\right ) \left (\frac {x (x+4) \left (x^2+2 y(x)\right )}{\sqrt [3]{x^3 (x+4)^3} \left (x^2-y(x)\right )}+2\right ) \left (\left (1-\frac {x (x+4) \left (x^2+2 y(x)\right )}{\sqrt [3]{x^3 (x+4)^3} \left (x^2-y(x)\right )}\right ) \log \left (2^{2/3} \left (1-\frac {x (x+4) \left (x^2+2 y(x)\right )}{\sqrt [3]{x^3 (x+4)^3} \left (x^2-y(x)\right )}\right )\right )+\left (\frac {x (x+4) \left (x^2+2 y(x)\right )}{\sqrt [3]{x^3 (x+4)^3} \left (x^2-y(x)\right )}-1\right ) \log \left (2^{2/3} \left (\frac {x (x+4) \left (x^2+2 y(x)\right )}{\sqrt [3]{x^3 (x+4)^3} \left (x^2-y(x)\right )}+2\right )\right )-3\right )}{27 \left (3 \left (2 \sqrt [3]{x^3 (x+4)^3}+x^2+4 x\right ) x^2 y(x)^2+2 \left (\sqrt [3]{x^3 (x+4)^3}-x^2-4 x\right ) y(x)^3+\left (\sqrt [3]{x^3 (x+4)^3}-x^2-4 x\right ) x^6\right )}=\frac {x (x+4) (x+4 \log (x))}{9 \sqrt [3]{2} \sqrt [3]{x^3 (x+4)^3}}+c_1,y(x)\right ] \]

✗ Sympy

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

Timed Out