Internal problem ID [10093]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-1 Equation of form
\(y''+f(x)y'+g(x)y=0\)
Problem number: 12.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+y^{\prime } a +\left (b x +c \right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.035 (sec). Leaf size: 53
dsolve(diff(y(x),x$2)+a*diff(y(x),x)+(b*x+c)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} {\mathrm e}^{-\frac {x a}{2}} \AiryAi \left (\frac {a^{2}-4 b x -4 c}{4 b^{\frac {2}{3}}}\right )+c_{2} {\mathrm e}^{-\frac {x a}{2}} \AiryBi \left (\frac {a^{2}-4 b x -4 c}{4 b^{\frac {2}{3}}}\right ) \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 67
DSolve[y''[x]+a*y'[x]+(b*x+c)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{-\frac {a x}{2}} \left (c_1 \text {Ai}\left (\frac {a^2-4 (c+b x)}{4 (-b)^{2/3}}\right )+c_2 \text {Bi}\left (\frac {a^2-4 (c+b x)}{4 (-b)^{2/3}}\right )\right ) \\ \end{align*}