Internal problem ID [11019]
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: 185.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{3} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+y c=0} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 63
dsolve(x^3*diff(y(x),x$2)+(a*x^2+b*x)*diff(y(x),x)+c*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} x^{1-a} \operatorname {KummerM}\left (\frac {b \left (a -1\right )-c}{b}, a , \frac {b}{x}\right )+c_{2} x^{1-a} \operatorname {KummerU}\left (\frac {b \left (a -1\right )-c}{b}, a , \frac {b}{x}\right ) \]
✓ Solution by Mathematica
Time used: 0.435 (sec). Leaf size: 62
DSolve[x^3*y''[x]+(a*x^2+b*x)*y'[x]+c*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 \operatorname {Hypergeometric1F1}\left (-\frac {c}{b},2-a,\frac {b}{x}\right )-(-1)^a c_2 b^{a-1} \left (\frac {1}{x}\right )^{a-1} \operatorname {Hypergeometric1F1}\left (a-\frac {b+c}{b},a,\frac {b}{x}\right ) \]