60.3.290 problem 1295

Internal problem ID [11300]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1295
Date solved : Tuesday, January 28, 2025 at 05:58:03 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.194 (sec). Leaf size: 102

dsolve(a*x^2*diff(diff(y(x),x),x)+b*x*diff(y(x),x)+(c*x^2+d*x+f)*y(x)=0,y(x), singsol=all)
 

\[ y = x^{-\frac {b}{2 a}} \left (\operatorname {WhittakerM}\left (-\frac {i d}{2 \sqrt {a}\, \sqrt {c}}, \frac {\sqrt {a^{2}+\left (-2 b -4 f \right ) a +b^{2}}}{2 a}, \frac {2 i \sqrt {c}\, x}{\sqrt {a}}\right ) c_{1} +\operatorname {WhittakerW}\left (-\frac {i d}{2 \sqrt {a}\, \sqrt {c}}, \frac {\sqrt {a^{2}+\left (-2 b -4 f \right ) a +b^{2}}}{2 a}, \frac {2 i \sqrt {c}\, x}{\sqrt {a}}\right ) c_{2} \right ) \]

✓ Solution by Mathematica

Time used: 0.388 (sec). Leaf size: 239

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

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