60.3.36 problem 1037

Internal problem ID [11046]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1037
Date solved : Tuesday, January 28, 2025 at 05:40:41 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.144 (sec). Leaf size: 64

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

\[ y = {\mathrm e}^{-\frac {x \left (b x +a \right )}{2}} x \left (\operatorname {KummerM}\left (\frac {a^{2}+12 b +4 c}{16 b}, \frac {3}{2}, b \,x^{2}\right ) c_{1} +\operatorname {KummerU}\left (\frac {a^{2}+12 b +4 c}{16 b}, \frac {3}{2}, b \,x^{2}\right ) c_{2} \right ) \]

✓ Solution by Mathematica

Time used: 0.047 (sec). Leaf size: 74

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

\[ y(x)\to e^{-\frac {1}{2} x (a+b x)} \left (c_1 \operatorname {HermiteH}\left (-\frac {a^2+4 (b+c)}{8 b},\sqrt {b} x\right )+c_2 \operatorname {Hypergeometric1F1}\left (\frac {a^2+4 (b+c)}{16 b},\frac {1}{2},b x^2\right )\right ) \]