60.3.206 problem 1210

Internal problem ID [11216]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1210
Date solved : Tuesday, January 28, 2025 at 05:41:55 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-2 x \left (x^{2}-a \right ) y^{\prime }+\left (2 n \,x^{2}+\left (\left (-1\right )^{n}-1\right ) a \right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 5.717 (sec). Leaf size: 81

dsolve(x^2*diff(diff(y(x),x),x)-2*x*(x^2-a)*diff(y(x),x)+(2*n*x^2+((-1)^n-1)*a)*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.348 (sec). Leaf size: 231

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

\[ y(x)\to i^{-a} (-1)^{\frac {1}{4} \left (1-\sqrt {4 a^2-4 a (-1)^n+1}\right )} x^{\frac {1}{2} \left (-\sqrt {4 a^2-4 a (-1)^n+1}-2 a+1\right )} \left (c_1 \operatorname {Hypergeometric1F1}\left (\frac {1}{4} \left (-2 a-2 n-\sqrt {4 a^2-4 (-1)^n a+1}+1\right ),1-\frac {1}{2} \sqrt {4 a^2-4 (-1)^n a+1},x^2\right )+c_2 i^{\sqrt {4 a^2-4 a (-1)^n+1}} x^{\sqrt {4 a^2-4 a (-1)^n+1}} \operatorname {Hypergeometric1F1}\left (\frac {1}{4} \left (-2 a-2 n+\sqrt {4 a^2-4 (-1)^n a+1}+1\right ),\frac {1}{2} \left (\sqrt {4 a^2-4 (-1)^n a+1}+2\right ),x^2\right )\right ) \]