Internal problem ID [11020]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-6 Equation of form
\((a_3 x^3+a_2 x^2 x+a_1 x+a_0) y''+f(x)y'+g(x)y=0\)
Problem number: 186.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{3} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+\left (x c +d \right ) y=0} \]
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 153
dsolve(x^3*diff(y(x),x$2)+(a*x^2+b*x)*diff(y(x),x)+(c*x+d)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} x^{-\frac {\sqrt {a^{2}-2 a -4 c +1}}{2}-\frac {a}{2}+\frac {1}{2}} \operatorname {KummerM}\left (\frac {\sqrt {a^{2}-2 a -4 c +1}\, b +b \left (a -1\right )-2 d}{2 b}, 1+\sqrt {a^{2}-2 a -4 c +1}, \frac {b}{x}\right )+c_{2} x^{-\frac {\sqrt {a^{2}-2 a -4 c +1}}{2}-\frac {a}{2}+\frac {1}{2}} \operatorname {KummerU}\left (\frac {\sqrt {a^{2}-2 a -4 c +1}\, b +b \left (a -1\right )-2 d}{2 b}, 1+\sqrt {a^{2}-2 a -4 c +1}, \frac {b}{x}\right ) \]
✓ Solution by Mathematica
Time used: 0.641 (sec). Leaf size: 255
DSolve[x^3*y''[x]+(a*x^2+b*x)*y'[x]+(c*x+d)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -i^{-\sqrt {a^2-2 a-4 c+1}+a+1} b^{\frac {1}{2} \left (-\sqrt {a^2-2 a-4 c+1}+a-1\right )} \left (\frac {1}{x}\right )^{\frac {1}{2} \left (-\sqrt {a^2-2 a-4 c+1}+a-1\right )} \left (c_2 i^{2 \sqrt {a^2-2 a-4 c+1}} b^{\sqrt {a^2-2 a-4 c+1}} \left (\frac {1}{x}\right )^{\sqrt {a^2-2 a-4 c+1}} \operatorname {Hypergeometric1F1}\left (\frac {1}{2} \left (a-\frac {2 d}{b}+\sqrt {a^2-2 a-4 c+1}-1\right ),\sqrt {a^2-2 a-4 c+1}+1,\frac {b}{x}\right )+c_1 \operatorname {Hypergeometric1F1}\left (\frac {1}{2} \left (a-\frac {2 d}{b}-\sqrt {a^2-2 a-4 c+1}-1\right ),1-\sqrt {a^2-2 a-4 c+1},\frac {b}{x}\right )\right ) \]