54.1.216 problem 221

Internal problem ID [11530]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 221
Date solved : Tuesday, September 30, 2025 at 08:57:16 PM
CAS classification : [[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

\begin{align*} \left (2 y+x +1\right ) y^{\prime }-2 y-x +1&=0 \end{align*}

✓ Maple. Time used: 0.020 (sec). Leaf size: 21

ode:=(2*y(x)+x+1)*diff(y(x),x)-2*y(x)-x+1 = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {x}{2}+\frac {2 \operatorname {LambertW}\left (\frac {c_1 \,{\mathrm e}^{-\frac {1}{4}+\frac {9 x}{4}}}{4}\right )}{3}+\frac {1}{6} \]

✓ Mathematica. Time used: 2.538 (sec). Leaf size: 43

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

\begin{align*} y(x)&\to \frac {1}{6} \left (4 W\left (-e^{\frac {9 x}{4}-1+c_1}\right )-3 x+1\right )\\ y(x)&\to \frac {1}{6} (1-3 x) \end{align*}

✓ Sympy. Time used: 2.545 (sec). Leaf size: 129

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

\[ \left [ y{\left (x \right )} = - \frac {x}{2} + \frac {2 W\left (- \frac {\sqrt [4]{C_{1} e^{9 x}}}{4 e^{\frac {1}{4}}}\right )}{3} + \frac {1}{6}, \ y{\left (x \right )} = - \frac {x}{2} + \frac {2 W\left (\frac {\sqrt [4]{C_{1} e^{9 x}}}{4 e^{\frac {1}{4}}}\right )}{3} + \frac {1}{6}, \ y{\left (x \right )} = - \frac {x}{2} + \frac {2 W\left (- \frac {i \sqrt [4]{C_{1} e^{9 x}}}{4 e^{\frac {1}{4}}}\right )}{3} + \frac {1}{6}, \ y{\left (x \right )} = - \frac {x}{2} + \frac {2 W\left (\frac {i \sqrt [4]{C_{1} e^{9 x}}}{4 e^{\frac {1}{4}}}\right )}{3} + \frac {1}{6}\right ] \]