Internal problem ID [9471]
Internal file name [OUTPUT/8411_Monday_June_06_2022_02_59_08_AM_46512108/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1142.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_2nd_order, _with_linear_symmetries]]
Unable to solve or complete the solution.
\[ \boxed {5 \left (a x +b \right ) y^{\prime \prime }+8 y^{\prime } a +c \left (a x +b \right )^{\frac {1}{5}} y=0} \]
Maple trace Kovacic algorithm successful
`Methods for second order ODEs: --- Trying classification methods --- trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying an equivalence, under non-integer power transformations, to LODEs admitting Liouvillian solutions. -> Trying a Liouvillian solution using Kovacics algorithm A Liouvillian solution exists Group is reducible or imprimitive <- Kovacics algorithm successful <- Equivalence, under non-integer power transformations successful`
✓ Solution by Maple
Time used: 0.281 (sec). Leaf size: 59
dsolve(5*(a*x+b)*diff(diff(y(x),x),x)+8*a*diff(y(x),x)+c*(a*x+b)^(1/5)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {c_{1} \sinh \left (\frac {\left (a x +b \right )^{\frac {3}{5}} \sqrt {5}\, \sqrt {-c}}{3 a}\right )+c_{2} \cosh \left (\frac {\left (a x +b \right )^{\frac {3}{5}} \sqrt {5}\, \sqrt {-c}}{3 a}\right )}{\left (a x +b \right )^{\frac {3}{5}}} \]
✓ Solution by Mathematica
Time used: 0.099 (sec). Leaf size: 89
DSolve[c*(b + a*x)^(1/5)*y[x] + 8*a*y'[x] + 5*(b + a*x)*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {3 a \left (2 c_1 \cos \left (\frac {\sqrt {5} \sqrt {c} (a x+b)^{3/5}}{3 a}\right )+c_2 \sin \left (\frac {\sqrt {5} \sqrt {c} (a x+b)^{3/5}}{3 a}\right )\right )}{\sqrt {5} \sqrt {c} (a x+b)^{3/5}} \]