60.3.203 problem 1207

Internal problem ID [11213]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1207
Date solved : Tuesday, January 28, 2025 at 05:41:53 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+\left (a x +b \right ) y^{\prime } x +\left (\operatorname {a1} \,x^{2}+\operatorname {b1} x +\operatorname {c1} \right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.165 (sec). Leaf size: 110

dsolve(x^2*diff(diff(y(x),x),x)+(a*x+b)*diff(y(x),x)*x+(a1*x^2+b1*x+c1)*y(x)=0,y(x), singsol=all)
 

\[ y = {\mathrm e}^{-\frac {a x}{2}} x^{-\frac {b}{2}} \left (c_{1} \operatorname {WhittakerM}\left (-\frac {a b -2 \operatorname {b1}}{2 \sqrt {a^{2}-4 \operatorname {a1}}}, \frac {\sqrt {b^{2}-2 b -4 \operatorname {c1} +1}}{2}, \sqrt {a^{2}-4 \operatorname {a1}}\, x \right )+c_{2} \operatorname {WhittakerW}\left (-\frac {a b -2 \operatorname {b1}}{2 \sqrt {a^{2}-4 \operatorname {a1}}}, \frac {\sqrt {b^{2}-2 b -4 \operatorname {c1} +1}}{2}, \sqrt {a^{2}-4 \operatorname {a1}}\, x \right )\right ) \]

✓ Solution by Mathematica

Time used: 0.797 (sec). Leaf size: 232

DSolve[(c1 + b1*x + a1*x^2)*y[x] + x*(b + a*x)*D[y[x],x] + x^2*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \left (c_1 \operatorname {HypergeometricU}\left (\frac {a b-2 \text {b1}+\sqrt {a^2-4 \text {a1}} \left (\sqrt {b^2-2 b-4 \text {c1}+1}+1\right )}{2 \sqrt {a^2-4 \text {a1}}},\sqrt {b^2-2 b-4 \text {c1}+1}+1,\sqrt {a^2-4 \text {a1}} x\right )+c_2 L_{\frac {-a b+2 \text {b1}-\sqrt {a^2-4 \text {a1}} \left (\sqrt {b^2-2 b-4 \text {c1}+1}+1\right )}{2 \sqrt {a^2-4 \text {a1}}}}^{\sqrt {b^2-2 b-4 \text {c1}+1}}\left (\sqrt {a^2-4 \text {a1}} x\right )\right ) \exp \left (\int _1^x-\frac {b+a K[1]+\sqrt {a^2-4 \text {a1}} K[1]-\sqrt {b^2-2 b-4 \text {c1}+1}-1}{2 K[1]}dK[1]\right ) \]