Internal problem ID [9033]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1454.
ODE order: 3.
ODE degree: 1.
CAS Maple gives this as type [[_3rd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime \prime }+2 a x y^{\prime }+a y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.006 (sec). Leaf size: 55
dsolve(diff(diff(diff(y(x),x),x),x)+2*a*x*diff(y(x),x)+a*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \AiryAi \left (-\frac {2^{\frac {2}{3}} a^{\frac {1}{3}} x}{2}\right )^{2}+c_{2} \AiryBi \left (-\frac {2^{\frac {2}{3}} a^{\frac {1}{3}} x}{2}\right )^{2}+c_{3} \AiryAi \left (-\frac {2^{\frac {2}{3}} a^{\frac {1}{3}} x}{2}\right ) \AiryBi \left (-\frac {2^{\frac {2}{3}} a^{\frac {1}{3}} x}{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 79
DSolve[a*y[x] + 2*a*x*y'[x] + Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 \text {Ai}\left (\sqrt [3]{-\frac {1}{2}} \sqrt [3]{a} x\right )^2+c_3 \text {Bi}\left (\sqrt [3]{-\frac {1}{2}} \sqrt [3]{a} x\right )^2+c_2 \text {Ai}\left (\sqrt [3]{-\frac {1}{2}} \sqrt [3]{a} x\right ) \text {Bi}\left (\sqrt [3]{-\frac {1}{2}} \sqrt [3]{a} x\right ) \\ \end{align*}