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
:
Tuesday, January 28, 2025 at 04:15:34 PM
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*}
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 470
dsolve(diff(y(x),x) + (3*a*x^2 + 4*a^2*x + b)*y(x)^3 + 3*x*y(x)^2=0,y(x), singsol=all)
\[
\frac {a \sqrt {3}\, \left (\operatorname {BesselI}\left (1+\frac {\sqrt {\frac {4 a^{3}-3 b}{a^{3}}}}{2}, -\frac {\sqrt {3}\, \sqrt {\frac {\left (4 a^{2} x +3 a \,x^{2}+b \right ) y-2 a}{a^{3} y}}}{2}\right ) c_{1} -\operatorname {BesselK}\left (1+\frac {\sqrt {\frac {4 a^{3}-3 b}{a^{3}}}}{2}, -\frac {\sqrt {3}\, \sqrt {\frac {\left (4 a^{2} x +3 a \,x^{2}+b \right ) y-2 a}{a^{3} y}}}{2}\right )\right ) \sqrt {\frac {\left (4 a^{2} x +3 a \,x^{2}+b \right ) y-2 a}{a^{3} y}}-\left (c_{1} \operatorname {BesselI}\left (\frac {\sqrt {\frac {4 a^{3}-3 b}{a^{3}}}}{2}, -\frac {\sqrt {3}\, \sqrt {\frac {\left (4 a^{2} x +3 a \,x^{2}+b \right ) y-2 a}{a^{3} y}}}{2}\right )+\operatorname {BesselK}\left (\frac {\sqrt {\frac {4 a^{3}-3 b}{a^{3}}}}{2}, -\frac {\sqrt {3}\, \sqrt {\frac {\left (4 a^{2} x +3 a \,x^{2}+b \right ) y-2 a}{a^{3} y}}}{2}\right )\right ) \left (\sqrt {\frac {4 a^{3}-3 b}{a^{3}}}\, a -2 a -3 x \right )}{\operatorname {BesselI}\left (1+\frac {\sqrt {\frac {4 a^{3}-3 b}{a^{3}}}}{2}, -\frac {\sqrt {3}\, \sqrt {\frac {\left (4 a^{2} x +3 a \,x^{2}+b \right ) y-2 a}{a^{3} y}}}{2}\right ) \sqrt {3}\, \sqrt {\frac {\left (4 a^{2} x +3 a \,x^{2}+b \right ) y-2 a}{a^{3} y}}\, a -\operatorname {BesselI}\left (\frac {\sqrt {\frac {4 a^{3}-3 b}{a^{3}}}}{2}, -\frac {\sqrt {3}\, \sqrt {\frac {\left (4 a^{2} x +3 a \,x^{2}+b \right ) y-2 a}{a^{3} y}}}{2}\right ) \left (\sqrt {\frac {4 a^{3}-3 b}{a^{3}}}\, a -2 a -3 x \right )} = 0
\]
✓ Solution by Mathematica
Time used: 4.072 (sec). Leaf size: 490
DSolve[D[y[x],x] + (3*a*x^2 + 4*a^2*x + b)*y[x]^3 + 3*x*y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [c_1=-\frac {i \sqrt {-\frac {4 a^3-3 b}{4 a^3}-\frac {3}{2 a^2 y(x)}+\frac {(-2 a-3 x)^2}{4 a^2}} \operatorname {BesselJ}\left (\frac {1}{2} \sqrt {\frac {4 a^3-3 b}{a^3}}+1,-i \sqrt {\frac {(-2 a-3 x)^2}{4 a^2}-\frac {4 a^3-3 b}{4 a^3}-\frac {3}{2 a^2 y(x)}}\right )+\left (\frac {1}{2} \sqrt {\frac {4 a^3-3 b}{a^3}}+\frac {-2 a-3 x}{2 a}\right ) \operatorname {BesselJ}\left (\frac {1}{2} \sqrt {\frac {4 a^3-3 b}{a^3}},-i \sqrt {\frac {(-2 a-3 x)^2}{4 a^2}-\frac {4 a^3-3 b}{4 a^3}-\frac {3}{2 a^2 y(x)}}\right )}{i \sqrt {-\frac {4 a^3-3 b}{4 a^3}-\frac {3}{2 a^2 y(x)}+\frac {(-2 a-3 x)^2}{4 a^2}} \operatorname {BesselY}\left (\frac {1}{2} \sqrt {\frac {4 a^3-3 b}{a^3}}+1,-i \sqrt {\frac {(-2 a-3 x)^2}{4 a^2}-\frac {4 a^3-3 b}{4 a^3}-\frac {3}{2 a^2 y(x)}}\right )+\left (\frac {1}{2} \sqrt {\frac {4 a^3-3 b}{a^3}}+\frac {-2 a-3 x}{2 a}\right ) \operatorname {BesselY}\left (\frac {1}{2} \sqrt {\frac {4 a^3-3 b}{a^3}},-i \sqrt {\frac {(-2 a-3 x)^2}{4 a^2}-\frac {4 a^3-3 b}{4 a^3}-\frac {3}{2 a^2 y(x)}}\right )},y(x)\right ]
\]