60.3.116 problem 1120

Internal problem ID [11126]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1120
Date solved : Tuesday, January 28, 2025 at 05:41:21 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.140 (sec). Leaf size: 109

dsolve(x*diff(diff(y(x),x),x)+(a*x+b)*diff(y(x),x)+(c*x+d)*y(x)=0,y(x), singsol=all)
 

\[ y = {\mathrm e}^{-\frac {x \left (\sqrt {a^{2}-4 c}+a \right )}{2}} \left (\operatorname {KummerU}\left (\frac {b \sqrt {a^{2}-4 c}+a b -2 d}{2 \sqrt {a^{2}-4 c}}, b , \sqrt {a^{2}-4 c}\, x \right ) c_{2} +\operatorname {KummerM}\left (\frac {b \sqrt {a^{2}-4 c}+a b -2 d}{2 \sqrt {a^{2}-4 c}}, b , \sqrt {a^{2}-4 c}\, x \right ) c_{1} \right ) \]

✓ Solution by Mathematica

Time used: 0.083 (sec). Leaf size: 135

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

\[ y(x)\to e^{-\frac {1}{2} x \left (\sqrt {a^2-4 c}+a\right )} \left (c_1 \operatorname {HypergeometricU}\left (\frac {a b+\sqrt {a^2-4 c} b-2 d}{2 \sqrt {a^2-4 c}},b,\sqrt {a^2-4 c} x\right )+c_2 L_{-\frac {a b+\sqrt {a^2-4 c} b-2 d}{2 \sqrt {a^2-4 c}}}^{b-1}\left (\sqrt {a^2-4 c} x\right )\right ) \]