Internal problem ID [10862]
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 Equation of form
\(y''+f(x)y'+g(x)y=0\)
Problem number: 28.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+\left (x c +d \right ) y=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 105
dsolve(diff(y(x),x$2)+(a*x+b)*diff(y(x),x)+(c*x+d)*y(x)=0,y(x), singsol=all)
\[ y = c_{1} {\mathrm e}^{-\frac {c x}{a}} \operatorname {KummerM}\left (\frac {a^{2} d -a b c +c^{2}}{2 a^{3}}, \frac {1}{2}, -\frac {\left (a^{2} x +a b -2 c \right )^{2}}{2 a^{3}}\right )+c_{2} {\mathrm e}^{-\frac {c x}{a}} \operatorname {KummerU}\left (\frac {a^{2} d -a b c +c^{2}}{2 a^{3}}, \frac {1}{2}, -\frac {\left (a^{2} x +a b -2 c \right )^{2}}{2 a^{3}}\right ) \]
✓ Solution by Mathematica
Time used: 0.116 (sec). Leaf size: 132
DSolve[y''[x]+(a*x+b)*y'[x]+(c*x+d)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{\frac {c x}{a}-\frac {a x^2}{2}-b x} \left (c_2 \operatorname {Hypergeometric1F1}\left (\frac {a^3-d a^2+b c a-c^2}{2 a^3},\frac {1}{2},\frac {\left (x a^2+b a-2 c\right )^2}{2 a^3}\right )+c_1 \operatorname {HermiteH}\left (\frac {-a^3+d a^2-b c a+c^2}{a^3},\frac {x a^2+b a-2 c}{\sqrt {2} a^{3/2}}\right )\right ) \]