Internal problem ID [9104]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1525.
ODE order: 3.
ODE degree: 1.
CAS Maple gives this as type [[_3rd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{6} y^{\prime \prime \prime }+6 x^{5} y^{\prime \prime }+a y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.019 (sec). Leaf size: 287
dsolve(x^6*diff(diff(diff(y(x),x),x),x)+6*x^5*diff(diff(y(x),x),x)+a*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} \left (-8 x^{3}+a \right )^{4} {\mathrm e}^{-\frac {\left (-a^{4}\right )^{\frac {1}{3}}}{a x}}}{\left (2 a x +\left (-a^{4}\right )^{\frac {1}{3}}\right )^{3} \left (4 a^{2} x^{2}-2 x \left (-a^{4}\right )^{\frac {1}{3}} a +\left (-a^{4}\right )^{\frac {2}{3}}\right )^{4}}+\frac {c_{2} \left (-8 x^{3}+a \right )^{4} {\mathrm e}^{-\frac {i \left (\sqrt {3}+i\right ) \left (-a^{4}\right )^{\frac {1}{3}}}{2 a x}}}{\left (\left (-a^{4}\right )^{\frac {1}{3}} \sqrt {3}-4 i a x +i \left (-a^{4}\right )^{\frac {1}{3}}\right )^{3} \left (4 i a x +\left (-a^{4}\right )^{\frac {1}{3}} \sqrt {3}-i \left (-a^{4}\right )^{\frac {1}{3}}\right )^{4} \left (2 a x +\left (-a^{4}\right )^{\frac {1}{3}}\right )^{4}}+\frac {c_{3} \left (-8 x^{3}+a \right )^{4} {\mathrm e}^{\frac {i \left (-i+\sqrt {3}\right ) \left (-a^{4}\right )^{\frac {1}{3}}}{2 a x}}}{\left (4 i a x +\left (-a^{4}\right )^{\frac {1}{3}} \sqrt {3}-i \left (-a^{4}\right )^{\frac {1}{3}}\right )^{3} \left (\left (-a^{4}\right )^{\frac {1}{3}} \sqrt {3}-4 i a x +i \left (-a^{4}\right )^{\frac {1}{3}}\right )^{4} \left (2 a x +\left (-a^{4}\right )^{\frac {1}{3}}\right )^{4}} \]
✓ Solution by Mathematica
Time used: 0.236 (sec). Leaf size: 101
DSolve[a*y[x] + 6*x^5*y''[x] + x^6*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 \left (-e^{\frac {\sqrt [3]{a}}{x}}\right ) \left (\sqrt [3]{a}-2 x\right )+c_2 e^{\frac {(-1)^{2/3} \sqrt [3]{a}}{x}} \left (x-\frac {1}{2} (-1)^{2/3} \sqrt [3]{a}\right )+c_3 e^{-\frac {\sqrt [3]{-1} \sqrt [3]{a}}{x}} \left (\frac {1}{2} \sqrt [3]{-1} \sqrt [3]{a}+x\right ) \\ \end{align*}