54.1.103 problem 105
Internal
problem
ID
[11417]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
105
Date
solved
:
Tuesday, September 30, 2025 at 08:16:08 PM
CAS
classification
:
[_rational, _Riccati]
\begin{align*} x y^{\prime }+a x y^{2}+b y+c x +d&=0 \end{align*}
✓ Maple. Time used: 0.031 (sec). Leaf size: 407
ode:=x*diff(y(x),x)+a*x*y(x)^2+b*y(x)+c*x+d = 0;
dsolve(ode,y(x), singsol=all);
\[
y = -\frac {4 c \left (a \,c^{3} \left (a d -b \sqrt {-a c}\right ) \operatorname {KummerM}\left (\frac {c b +\sqrt {-a c}\, d +2 c}{2 c}, b +1, 2 x \sqrt {-a c}\right )-\frac {c_1 \left (a \,d^{2}+b^{2} c \right ) \operatorname {KummerU}\left (\frac {c b +\sqrt {-a c}\, d +2 c}{2 c}, b +1, 2 x \sqrt {-a c}\right )}{4}+a \,c^{3} \left (b \sqrt {-a c}+a d \right ) \operatorname {KummerM}\left (\frac {\sqrt {-a c}\, d +c b}{2 c}, b +1, 2 x \sqrt {-a c}\right )-\frac {\operatorname {KummerU}\left (\frac {\sqrt {-a c}\, d +c b}{2 c}, b +1, 2 x \sqrt {-a c}\right ) c_1 \left (c b -\sqrt {-a c}\, d \right )}{2}\right )}{4 a^{2} c^{3} \left (\sqrt {-a c}\, d +c b \right ) \operatorname {KummerM}\left (\frac {c b +\sqrt {-a c}\, d +2 c}{2 c}, b +1, 2 x \sqrt {-a c}\right )-\sqrt {-a c}\, c_1 \left (a \,d^{2}+b^{2} c \right ) \operatorname {KummerU}\left (\frac {c b +\sqrt {-a c}\, d +2 c}{2 c}, b +1, 2 x \sqrt {-a c}\right )+4 c \left (a^{2} c^{2} \left (c b -\sqrt {-a c}\, d \right ) \operatorname {KummerM}\left (\frac {\sqrt {-a c}\, d +c b}{2 c}, b +1, 2 x \sqrt {-a c}\right )+\frac {\operatorname {KummerU}\left (\frac {\sqrt {-a c}\, d +c b}{2 c}, b +1, 2 x \sqrt {-a c}\right ) c_1 \left (b \sqrt {-a c}+a d \right )}{2}\right )}
\]
✓ Mathematica. Time used: 0.291 (sec). Leaf size: 541
ode=x*D[y[x],x] + a*x*y[x]^2 + b*y[x] + c*x + d==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to -\frac {i \left (\sqrt {c} c_1 \operatorname {HypergeometricU}\left (\frac {1}{2} \left (b+\frac {i \sqrt {a} d}{\sqrt {c}}\right ),b,2 i \sqrt {a} \sqrt {c} x\right )+c_1 \left (b \sqrt {c}+i \sqrt {a} d\right ) \operatorname {HypergeometricU}\left (\frac {1}{2} \left (b+\frac {i \sqrt {a} d}{\sqrt {c}}+2\right ),b+1,2 i \sqrt {a} \sqrt {c} x\right )+\sqrt {c} \left (2 L_{-\frac {b}{2}-\frac {i \sqrt {a} d}{2 \sqrt {c}}-1}^b\left (2 i \sqrt {a} \sqrt {c} x\right )+L_{-\frac {b}{2}-\frac {i \sqrt {a} d}{2 \sqrt {c}}}^{b-1}\left (2 i \sqrt {a} \sqrt {c} x\right )\right )\right )}{\sqrt {a} \left (c_1 \operatorname {HypergeometricU}\left (\frac {1}{2} \left (b+\frac {i \sqrt {a} d}{\sqrt {c}}\right ),b,2 i \sqrt {a} \sqrt {c} x\right )+L_{-\frac {b}{2}-\frac {i \sqrt {a} d}{2 \sqrt {c}}}^{b-1}\left (2 i \sqrt {a} \sqrt {c} x\right )\right )}\\ y(x)&\to \frac {\frac {\left (\sqrt {a} d-i b \sqrt {c}\right ) \operatorname {HypergeometricU}\left (\frac {1}{2} \left (b+\frac {i \sqrt {a} d}{\sqrt {c}}+2\right ),b+1,2 i \sqrt {a} \sqrt {c} x\right )}{\operatorname {HypergeometricU}\left (\frac {1}{2} \left (b+\frac {i \sqrt {a} d}{\sqrt {c}}\right ),b,2 i \sqrt {a} \sqrt {c} x\right )}-i \sqrt {c}}{\sqrt {a}}\\ y(x)&\to \frac {\frac {\left (\sqrt {a} d-i b \sqrt {c}\right ) \operatorname {HypergeometricU}\left (\frac {1}{2} \left (b+\frac {i \sqrt {a} d}{\sqrt {c}}+2\right ),b+1,2 i \sqrt {a} \sqrt {c} x\right )}{\operatorname {HypergeometricU}\left (\frac {1}{2} \left (b+\frac {i \sqrt {a} d}{\sqrt {c}}\right ),b,2 i \sqrt {a} \sqrt {c} x\right )}-i \sqrt {c}}{\sqrt {a}} \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
c = symbols("c")
d = symbols("d")
y = Function("y")
ode = Eq(a*x*y(x)**2 + b*y(x) + c*x + d + x*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
RecursionError : maximum recursion depth exceeded