61.31.4 problem 185

Internal problem ID [12685]
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
Problem number : 185
Date solved : Tuesday, January 28, 2025 at 08:10:58 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{3} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+c y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.271 (sec). Leaf size: 146

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)
 

\[ y = \frac {x^{-a} \left (-c_{2} x c \left (a b -c \right ) \left (-c +b \right ) \operatorname {KummerU}\left (\frac {\left (a +1\right ) b -c}{b}, a , \frac {b}{x}\right )+\left (x c_{1} b \left (a b -c \right ) \operatorname {KummerM}\left (\frac {\left (a +1\right ) b -c}{b}, a , \frac {b}{x}\right )-\left (b c_{1} \operatorname {KummerM}\left (\frac {a b -c}{b}, a , \frac {b}{x}\right )-c c_{2} \operatorname {KummerU}\left (\frac {a b -c}{b}, a , \frac {b}{x}\right )\right ) \left (a x b +b^{2}-2 c x \right )\right ) b \right )}{b^{2} c} \]

✓ Solution by Mathematica

Time used: 0.278 (sec). Leaf size: 62

DSolve[x^3*D[y[x],{x,2}]+(a*x^2+b*x)*D[y[x],x]+c*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_1 \operatorname {Hypergeometric1F1}\left (-\frac {c}{b},2-a,\frac {b}{x}\right )-(-1)^a c_2 b^{a-1} \left (\frac {1}{x}\right )^{a-1} \operatorname {Hypergeometric1F1}\left (a-\frac {b+c}{b},a,\frac {b}{x}\right ) \]