82.54.6 problem Ex. 6

Internal problem ID [19022]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter IX. Equations of the second order. problems at end of chapter at page 120
Problem number : Ex. 6
Date solved : Tuesday, January 28, 2025 at 08:29:25 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-2 b y^{\prime }+b^{2} x^{2} y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.045 (sec). Leaf size: 47

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

\[ y \left (x \right ) = {\mathrm e}^{-\frac {b x \left (i x -2\right )}{2}} x \left (\operatorname {KummerM}\left (\frac {3}{4}-\frac {i b}{4}, \frac {3}{2}, i b \,x^{2}\right ) c_{1} +\operatorname {KummerU}\left (\frac {3}{4}-\frac {i b}{4}, \frac {3}{2}, i b \,x^{2}\right ) c_{2} \right ) \]

✓ Solution by Mathematica

Time used: 0.040 (sec). Leaf size: 75

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

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