60.3.59 problem 1060

Internal problem ID [11069]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1060
Date solved : Tuesday, January 28, 2025 at 05:40:51 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+a \,x^{q -1} y^{\prime }+b \,x^{q -2} y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.901 (sec). Leaf size: 71

dsolve(diff(diff(y(x),x),x)+a*x^(q-1)*diff(y(x),x)+b*x^(q-2)*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.062 (sec). Leaf size: 81

DSolve[b*x^(-2 + q)*y[x] + a*x^(-1 + q)*D[y[x],x] + D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

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