Internal problem ID [9125]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 4, linear fourth order
Problem number: 1546.
ODE order: 4.
ODE degree: 1.
CAS Maple gives this as type [[_high_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime \prime \prime }+4 a x y^{\prime \prime \prime }+6 a^{2} x^{2} y^{\prime \prime }+4 a^{3} x^{3} y^{\prime }+a^{4} x^{4} y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.019 (sec). Leaf size: 81
dsolve(diff(diff(diff(diff(y(x),x),x),x),x)+4*a*x*diff(diff(diff(y(x),x),x),x)+6*a^2*x^2*diff(diff(y(x),x),x)+4*a^3*x^3*diff(y(x),x)+a^4*x^4*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = {\mathrm e}^{-\frac {a \,x^{2}}{2}} \left (c_{1} {\mathrm e}^{-\sqrt {-a \sqrt {6}+3 a}\, x}+c_{2} {\mathrm e}^{\sqrt {-a \sqrt {6}+3 a}\, x}+c_{3} {\mathrm e}^{-\sqrt {a \sqrt {6}+3 a}\, x}+c_{4} {\mathrm e}^{\sqrt {a \sqrt {6}+3 a}\, x}\right ) \]
✓ Solution by Mathematica
Time used: 0.414 (sec). Leaf size: 162
DSolve[a^4*x^4*y[x] + 4*a^3*x^3*y'[x] + 6*a^2*x^2*y''[x] + 4*a*x*Derivative[3][y][x] + Derivative[4][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {e^{-\frac {a x^2}{2}-\sqrt {3+\sqrt {6}} \sqrt {a} x} \left (6 a \left (c_1 e^{\sqrt {6-2 \sqrt {3}} \sqrt {a} x}+c_2 e^{\sqrt {2 \left (3+\sqrt {3}\right )} \sqrt {a} x}\right )+\sqrt {6} \sqrt {-\left (\left (\sqrt {6}-3\right ) a\right )} c_4 e^{\frac {2 a x}{\sqrt {a-\sqrt {\frac {2}{3}} a}}}+\sqrt {18-6 \sqrt {6}} \sqrt {a} c_3\right )}{6 a} \\ \end{align*}