61.27.18 problem 28

Internal problem ID [12528]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-2
Problem number : 28
Date solved : Tuesday, January 28, 2025 at 08:02:07 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.092 (sec). Leaf size: 98

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

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

✓ Solution by Mathematica

Time used: 0.078 (sec). Leaf size: 132

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

\[ y(x)\to e^{\frac {c x}{a}-\frac {a x^2}{2}-b x} \left (c_2 \operatorname {Hypergeometric1F1}\left (\frac {a^3-d a^2+b c a-c^2}{2 a^3},\frac {1}{2},\frac {\left (x a^2+b a-2 c\right )^2}{2 a^3}\right )+c_1 \operatorname {HermiteH}\left (\frac {-a^3+d a^2-b c a+c^2}{a^3},\frac {x a^2+b a-2 c}{\sqrt {2} a^{3/2}}\right )\right ) \]