Internal problem ID [9130]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 4, linear fourth order
Problem number: 1551.
ODE order: 4.
ODE degree: 1.
CAS Maple gives this as type [[_high_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime \prime \prime }-2 \left (\nu ^{2} x^{2}+6\right ) y^{\prime \prime }+\nu ^{2} \left (\nu ^{2} x^{2}+4\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 63
dsolve(x^2*diff(diff(diff(diff(y(x),x),x),x),x)-2*(nu^2*x^2+6)*diff(diff(y(x),x),x)+nu^2*(nu^2*x^2+4)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} {\mathrm e}^{\nu x}}{x}+\frac {c_{2} {\mathrm e}^{-\nu x}}{x}+c_{3} {\mathrm e}^{\nu x} \left (\nu ^{2} x^{2}-6 \nu x +15\right )+c_{4} {\mathrm e}^{-\nu x} \left (\nu ^{2} x^{2}+6 \nu x +15\right ) \]
✓ Solution by Mathematica
Time used: 0.2 (sec). Leaf size: 78
DSolve[nu^2*(4 + nu^2*x^2)*y[x] - 2*(6 + nu^2*x^2)*y''[x] + x^2*Derivative[4][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {e^{-\nu x} \left (c_3 \left (\nu \left (-\nu x^3+\nu -6 x^2+6\right )-15 (x-1)\right )+e^{2 \nu x} \left (c_4 \left (\nu \left (-\nu x^3+\nu +6 x^2-6\right )-15 (x-1)\right )+c_2\right )+c_1\right )}{x} \\ \end{align*}