60.3.186 problem 1190

Internal problem ID [11196]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1190
Date solved : Tuesday, January 28, 2025 at 05:41:47 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.110 (sec). Leaf size: 38

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

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

✓ Solution by Mathematica

Time used: 0.761 (sec). Leaf size: 105

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

\[ y(x)\to \left (c_1 \operatorname {HypergeometricU}\left (\frac {1}{2} \left (-2 a+\sqrt {1-4 b}+1\right ),\sqrt {1-4 b}+1,x\right )+c_2 L_{a-\frac {1}{2} \sqrt {1-4 b}-\frac {1}{2}}^{\sqrt {1-4 b}}(x)\right ) \exp \left (\int _1^x\frac {-2 K[1]+\sqrt {1-4 b}+1}{2 K[1]}dK[1]\right ) \]