60.1.43 problem 43
Internal
problem
ID
[10057]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
43
Date
solved
:
Friday, March 14, 2025 at 01:51:25 AM
CAS
classification
:
[_Abel]
\begin{align*} y^{\prime }+\left (4 a^{2} x +3 a \,x^{2}+b \right ) y^{3}+3 x y^{2}&=0 \end{align*}
✓ Maple. Time used: 0.002 (sec). Leaf size: 470
ode:=diff(y(x),x)+(4*a^2*x+3*a*x^2+b)*y(x)^3+3*x*y(x)^2 = 0;
dsolve(ode,y(x), singsol=all);
\[
\frac {a \sqrt {3}\, \left (\operatorname {BesselI}\left (1+\frac {\sqrt {\frac {4 a^{3}-3 b}{a^{3}}}}{2}, -\frac {\sqrt {3}\, \sqrt {\frac {\left (4 a^{2} x +3 a \,x^{2}+b \right ) y-2 a}{a^{3} y}}}{2}\right ) c_{1} -\operatorname {BesselK}\left (1+\frac {\sqrt {\frac {4 a^{3}-3 b}{a^{3}}}}{2}, -\frac {\sqrt {3}\, \sqrt {\frac {\left (4 a^{2} x +3 a \,x^{2}+b \right ) y-2 a}{a^{3} y}}}{2}\right )\right ) \sqrt {\frac {\left (4 a^{2} x +3 a \,x^{2}+b \right ) y-2 a}{a^{3} y}}-\left (c_{1} \operatorname {BesselI}\left (\frac {\sqrt {\frac {4 a^{3}-3 b}{a^{3}}}}{2}, -\frac {\sqrt {3}\, \sqrt {\frac {\left (4 a^{2} x +3 a \,x^{2}+b \right ) y-2 a}{a^{3} y}}}{2}\right )+\operatorname {BesselK}\left (\frac {\sqrt {\frac {4 a^{3}-3 b}{a^{3}}}}{2}, -\frac {\sqrt {3}\, \sqrt {\frac {\left (4 a^{2} x +3 a \,x^{2}+b \right ) y-2 a}{a^{3} y}}}{2}\right )\right ) \left (\sqrt {\frac {4 a^{3}-3 b}{a^{3}}}\, a -2 a -3 x \right )}{\operatorname {BesselI}\left (1+\frac {\sqrt {\frac {4 a^{3}-3 b}{a^{3}}}}{2}, -\frac {\sqrt {3}\, \sqrt {\frac {\left (4 a^{2} x +3 a \,x^{2}+b \right ) y-2 a}{a^{3} y}}}{2}\right ) \sqrt {3}\, \sqrt {\frac {\left (4 a^{2} x +3 a \,x^{2}+b \right ) y-2 a}{a^{3} y}}\, a -\operatorname {BesselI}\left (\frac {\sqrt {\frac {4 a^{3}-3 b}{a^{3}}}}{2}, -\frac {\sqrt {3}\, \sqrt {\frac {\left (4 a^{2} x +3 a \,x^{2}+b \right ) y-2 a}{a^{3} y}}}{2}\right ) \left (\sqrt {\frac {4 a^{3}-3 b}{a^{3}}}\, a -2 a -3 x \right )} = 0
\]
✓ Mathematica. Time used: 4.072 (sec). Leaf size: 490
ode=D[y[x],x] + (3*a*x^2 + 4*a^2*x + b)*y[x]^3 + 3*x*y[x]^2==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [c_1=-\frac {i \sqrt {-\frac {4 a^3-3 b}{4 a^3}-\frac {3}{2 a^2 y(x)}+\frac {(-2 a-3 x)^2}{4 a^2}} \operatorname {BesselJ}\left (\frac {1}{2} \sqrt {\frac {4 a^3-3 b}{a^3}}+1,-i \sqrt {\frac {(-2 a-3 x)^2}{4 a^2}-\frac {4 a^3-3 b}{4 a^3}-\frac {3}{2 a^2 y(x)}}\right )+\left (\frac {1}{2} \sqrt {\frac {4 a^3-3 b}{a^3}}+\frac {-2 a-3 x}{2 a}\right ) \operatorname {BesselJ}\left (\frac {1}{2} \sqrt {\frac {4 a^3-3 b}{a^3}},-i \sqrt {\frac {(-2 a-3 x)^2}{4 a^2}-\frac {4 a^3-3 b}{4 a^3}-\frac {3}{2 a^2 y(x)}}\right )}{i \sqrt {-\frac {4 a^3-3 b}{4 a^3}-\frac {3}{2 a^2 y(x)}+\frac {(-2 a-3 x)^2}{4 a^2}} \operatorname {BesselY}\left (\frac {1}{2} \sqrt {\frac {4 a^3-3 b}{a^3}}+1,-i \sqrt {\frac {(-2 a-3 x)^2}{4 a^2}-\frac {4 a^3-3 b}{4 a^3}-\frac {3}{2 a^2 y(x)}}\right )+\left (\frac {1}{2} \sqrt {\frac {4 a^3-3 b}{a^3}}+\frac {-2 a-3 x}{2 a}\right ) \operatorname {BesselY}\left (\frac {1}{2} \sqrt {\frac {4 a^3-3 b}{a^3}},-i \sqrt {\frac {(-2 a-3 x)^2}{4 a^2}-\frac {4 a^3-3 b}{4 a^3}-\frac {3}{2 a^2 y(x)}}\right )},y(x)\right ]
\]
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq(3*x*y(x)**2 + (4*a**2*x + 3*a*x**2 + b)*y(x)**3 + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out