Internal problem ID [7624]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 43.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Abel]
Solve \begin {gather*} \boxed {y^{\prime }+\left (4 a^{2} x +3 x^{2} a +b \right ) y^{3}+3 x y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 384
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)
\[ c_{1}+\frac {-\left (\frac {\sqrt {\frac {4 a^{3}-3 b}{a^{3}}}}{2}-\frac {2 a +3 x}{2 a}\right ) \BesselK \left (\frac {\sqrt {\frac {4 a^{3}-3 b}{a^{3}}}}{2}, -\frac {\sqrt {3}\, \sqrt {\frac {4 a^{2} x y \relax (x )+3 a \,x^{2} y \relax (x )+b y \relax (x )-2 a}{a^{3} y \relax (x )}}}{2}\right )-\frac {\BesselK \left (\frac {\sqrt {\frac {4 a^{3}-3 b}{a^{3}}}}{2}+1, -\frac {\sqrt {3}\, \sqrt {\frac {4 a^{2} x y \relax (x )+3 a \,x^{2} y \relax (x )+b y \relax (x )-2 a}{a^{3} y \relax (x )}}}{2}\right ) \sqrt {3}\, \sqrt {\frac {4 a^{2} x y \relax (x )+3 a \,x^{2} y \relax (x )+b y \relax (x )-2 a}{a^{3} y \relax (x )}}}{2}}{-\left (\frac {\sqrt {\frac {4 a^{3}-3 b}{a^{3}}}}{2}-\frac {2 a +3 x}{2 a}\right ) \BesselI \left (\frac {\sqrt {\frac {4 a^{3}-3 b}{a^{3}}}}{2}, -\frac {\sqrt {3}\, \sqrt {\frac {4 a^{2} x y \relax (x )+3 a \,x^{2} y \relax (x )+b y \relax (x )-2 a}{a^{3} y \relax (x )}}}{2}\right )+\frac {\BesselI \left (\frac {\sqrt {\frac {4 a^{3}-3 b}{a^{3}}}}{2}+1, -\frac {\sqrt {3}\, \sqrt {\frac {4 a^{2} x y \relax (x )+3 a \,x^{2} y \relax (x )+b y \relax (x )-2 a}{a^{3} y \relax (x )}}}{2}\right ) \sqrt {3}\, \sqrt {\frac {4 a^{2} x y \relax (x )+3 a \,x^{2} y \relax (x )+b y \relax (x )-2 a}{a^{3} y \relax (x )}}}{2}} = 0 \]
✓ Solution by Mathematica
Time used: 4.117 (sec). Leaf size: 490
DSolve[y'[x] + (3*a*x^2 + 4*a^2*x + b)*y[x]^3 + 3*x*y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\[ \operatorname {Solve}\left [c_1=-\frac {i \sqrt {-\frac {4 a^3-3 b}{4 a^3}-\frac {3}{2 a^2 y(x)}+\frac {(-2 a-3 x)^2}{4 a^2}} J_{\frac {1}{2} \sqrt {\frac {4 a^3-3 b}{a^3}}+1}\left (-i \sqrt {\frac {(-2 a-3 x)^2}{4 a^2}-\frac {4 a^3-3 b}{4 a^3}-\frac {3}{2 a^2 y(x)}}\right )+\left (\frac {1}{2} \sqrt {\frac {4 a^3-3 b}{a^3}}+\frac {-2 a-3 x}{2 a}\right ) J_{\frac {1}{2} \sqrt {\frac {4 a^3-3 b}{a^3}}}\left (-i \sqrt {\frac {(-2 a-3 x)^2}{4 a^2}-\frac {4 a^3-3 b}{4 a^3}-\frac {3}{2 a^2 y(x)}}\right )}{i \sqrt {-\frac {4 a^3-3 b}{4 a^3}-\frac {3}{2 a^2 y(x)}+\frac {(-2 a-3 x)^2}{4 a^2}} Y_{\frac {1}{2} \sqrt {\frac {4 a^3-3 b}{a^3}}+1}\left (-i \sqrt {\frac {(-2 a-3 x)^2}{4 a^2}-\frac {4 a^3-3 b}{4 a^3}-\frac {3}{2 a^2 y(x)}}\right )+\left (\frac {1}{2} \sqrt {\frac {4 a^3-3 b}{a^3}}+\frac {-2 a-3 x}{2 a}\right ) Y_{\frac {1}{2} \sqrt {\frac {4 a^3-3 b}{a^3}}}\left (-i \sqrt {\frac {(-2 a-3 x)^2}{4 a^2}-\frac {4 a^3-3 b}{4 a^3}-\frac {3}{2 a^2 y(x)}}\right )},y(x)\right ] \]