Internal problem ID [9878]
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]]
\[ \boxed {y^{\prime \prime \prime \prime }+4 a x y^{\prime \prime \prime }+6 a^{2} x^{2} y^{\prime \prime }+4 a^{3} x^{3} y^{\prime }+y a^{4} x^{4}=0} \]
✓ Solution by Maple
Time used: 0.016 (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)
\[ y \left (x \right ) = {\mathrm e}^{-\frac {a \,x^{2}}{2}} \left (c_{1} {\mathrm e}^{-\sqrt {-a \sqrt {6}+3 a}\, x}+c_{2} {\mathrm e}^{\sqrt {-a \sqrt {6}+3 a}\, x}+c_{3} {\mathrm e}^{-\sqrt {a \sqrt {6}+3 a}\, x}+c_{4} {\mathrm e}^{\sqrt {a \sqrt {6}+3 a}\, x}\right ) \]
✓ Solution by Mathematica
Time used: 0.737 (sec). Leaf size: 165
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]
\[ y(x)\to \frac {e^{-\frac {a x^2}{2}-\sqrt {3+\sqrt {6}} \sqrt {a} x} \left (6 a \left (c_1 e^{\frac {\left (-3+\sqrt {3}+\sqrt {6}\right ) a x}{\sqrt {-\left (\left (\sqrt {6}-3\right ) a\right )}}}+c_2 e^{\frac {\left (3+\sqrt {3}-\sqrt {6}\right ) a x}{\sqrt {-\left (\left (\sqrt {6}-3\right ) a\right )}}}\right )+\sqrt {6} \sqrt {-\left (\left (\sqrt {6}-3\right ) a\right )} \left (c_4 e^{\frac {2 a x}{\sqrt {a-\sqrt {\frac {2}{3}} a}}}+c_3\right )\right )}{6 a} \]