61.26.6 problem 6

Internal problem ID [12506]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2 Equations Containing Power Functions. page 213
Problem number : 6
Date solved : Tuesday, January 28, 2025 at 08:01:56 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.135 (sec). Leaf size: 142

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

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

✓ Solution by Mathematica

Time used: 0.058 (sec). Leaf size: 92

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

\[ y(x)\to c_2 \operatorname {ParabolicCylinderD}\left (-\frac {b^2 c^2}{8 a^{3/2}}-\frac {1}{2},\frac {i (b c+2 a x)}{\sqrt {2} a^{3/4}}\right )+c_1 \operatorname {ParabolicCylinderD}\left (\frac {1}{8} \left (\frac {b^2 c^2}{a^{3/2}}-4\right ),\frac {b c+2 a x}{\sqrt {2} a^{3/4}}\right ) \]