Internal problem ID [10968]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-4 Equation of form
\(x^2 y''+f(x)y'+g(x)y=0\)
Problem number: 134.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+y c=0} \]
✓ Solution by Maple
Time used: 0.062 (sec). Leaf size: 135
dsolve(x^2*diff(y(x),x$2)+(a*x+b)*diff(y(x),x)+c*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} x^{-\frac {\sqrt {a^{2}-2 a -4 c +1}}{2}-\frac {a}{2}+\frac {1}{2}} \operatorname {KummerM}\left (-\frac {1}{2}+\frac {\sqrt {a^{2}-2 a -4 c +1}}{2}+\frac {a}{2}, 1+\sqrt {a^{2}-2 a -4 c +1}, \frac {b}{x}\right )+c_{2} x^{-\frac {\sqrt {a^{2}-2 a -4 c +1}}{2}-\frac {a}{2}+\frac {1}{2}} \operatorname {KummerU}\left (-\frac {1}{2}+\frac {\sqrt {a^{2}-2 a -4 c +1}}{2}+\frac {a}{2}, 1+\sqrt {a^{2}-2 a -4 c +1}, \frac {b}{x}\right ) \]
✓ Solution by Mathematica
Time used: 0.574 (sec). Leaf size: 243
DSolve[x^2*y''[x]+(a*x+b)*y'[x]+c*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -i^{-\sqrt {a^2-2 a-4 c+1}+a+1} b^{\frac {1}{2} \left (-\sqrt {a^2-2 a-4 c+1}+a-1\right )} \left (\frac {1}{x}\right )^{\frac {1}{2} \left (-\sqrt {a^2-2 a-4 c+1}+a-1\right )} \left (c_2 i^{2 \sqrt {a^2-2 a-4 c+1}} b^{\sqrt {a^2-2 a-4 c+1}} \left (\frac {1}{x}\right )^{\sqrt {a^2-2 a-4 c+1}} \operatorname {Hypergeometric1F1}\left (\frac {1}{2} \left (a+\sqrt {a^2-2 a-4 c+1}-1\right ),\sqrt {a^2-2 a-4 c+1}+1,\frac {b}{x}\right )+c_1 \operatorname {Hypergeometric1F1}\left (\frac {1}{2} \left (a-\sqrt {a^2-2 a-4 c+1}-1\right ),1-\sqrt {a^2-2 a-4 c+1},\frac {b}{x}\right )\right ) \]