60.3.113 problem 1117

Internal problem ID [11123]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1117
Date solved : Tuesday, January 28, 2025 at 05:41:18 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }-\left (a +b \right ) \left (1+x \right ) y^{\prime }+a b x y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.106 (sec). Leaf size: 82

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

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

✓ Solution by Mathematica

Time used: 1.440 (sec). Leaf size: 98

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

\[ y(x)\to \left (c_1 \operatorname {HypergeometricU}\left (\frac {a^2+b a+a-b}{a-b},a+b+2,(a-b) x\right )+c_2 L_{-\frac {a^2+b a+a-b}{a-b}}^{a+b+1}((a-b) x)\right ) \exp \left (\int _1^x\frac {a+b+b K[1]+1}{K[1]}dK[1]\right ) \]