33.1.6 problem problem 169

Internal problem ID [6024]
Book : Differential Gleichungen, Kamke, 3rd ed, Abel ODEs
Section : Abel ODE with constant invariant
Problem number : problem 169
Date solved : Sunday, March 30, 2025 at 10:32:38 AM
CAS classification : [_rational, _Abel]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 153

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

\[ \frac {\left (\operatorname {erf}\left (\frac {\sqrt {2}\, \left (c y+a \left (a x +b \right )\right )}{2 \sqrt {a}\, y \left (a x +b \right )}\right ) {\mathrm e}^{\frac {\left (c y+a \left (a x +b \right )\right )^{2}}{2 y^{2} \left (a x +b \right )^{2} a}} \sqrt {\pi }\, \sqrt {2}\, a c +2 a^{{3}/{2}} \left (a x +b \right )\right ) {\mathrm e}^{-\frac {\left (\left (a x +b +c \right ) y+a \left (a x +b \right )\right ) \left (\left (-a x -b +c \right ) y+a \left (a x +b \right )\right )}{2 y^{2} \left (a x +b \right )^{2} a}}+2 c_1 \,a^{{5}/{2}}}{2 a^{{5}/{2}}} = 0 \]

✓ Mathematica. Time used: 1.356 (sec). Leaf size: 149

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

\[ \text {Solve}\left [-\frac {c}{\sqrt {-a (a x+b)^2}}=\frac {2 \exp \left (\frac {1}{2} \left (-\frac {c}{\sqrt {-a (a x+b)^2}}-\frac {\left (-a (a x+b)^2\right )^{3/2}}{a y(x) (a x+b)^3}\right )^2\right )}{\sqrt {2 \pi } \text {erfi}\left (\frac {-\frac {c}{\sqrt {-a (a x+b)^2}}-\frac {\left (-a (a x+b)^2\right )^{3/2}}{a y(x) (a x+b)^3}}{\sqrt {2}}\right )+2 c_1},y(x)\right ] \]

✗ Sympy

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

Timed Out