Internal problem ID [11310]
Internal file name [OUTPUT/10296_Wednesday_December_21_2022_03_48_01_PM_43515805/index.tex]
Book: An elementary treatise on differential equations by Abraham Cohen. DC heath publishers.
1906
Section: Chapter VIII, Linear differential equations of the second order. Article 55. Summary.
Page 129
Problem number: Ex 10.
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.
\[ \boxed {x^{4} y^{\prime \prime }+2 x^{3} \left (1+x \right ) y^{\prime }+y n^{2}=0} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying a Liouvillian solution using Kovacics algorithm <- No Liouvillian solutions exists -> Trying a solution in terms of special functions: -> Bessel -> elliptic -> Legendre -> Whittaker -> hyper3: Equivalence to 1F1 under a power @ Moebius -> hypergeometric -> heuristic approach -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius -> Mathieu -> Equivalence to the rational form of Mathieu ODE under a power @ Moebius trying a solution in terms of MeijerG functions -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius <- Heun successful: received ODE is equivalent to the HeunD ODE, case c = 0 `
✓ Solution by Maple
Time used: 0.422 (sec). Leaf size: 297
dsolve(x^4*diff(y(x),x$2)+2*x^3*(1+x)*diff(y(x),x)+n^2*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {c_{1} \operatorname {HeunD}\left (8 \left (-n^{2}\right )^{\frac {1}{4}}, \frac {-8 i \left (-n^{2}\right )^{\frac {3}{4}}-n +8 \sqrt {-n^{2}}\, n}{n}, -\frac {16 i \left (-n^{2}\right )^{\frac {3}{4}}}{n}, \frac {n -8 i \left (-n^{2}\right )^{\frac {3}{4}}-8 \sqrt {-n^{2}}\, n}{n}, \frac {\left (-n^{2}\right )^{\frac {1}{4}} x -i n}{\left (-n^{2}\right )^{\frac {1}{4}} x +i n}\right ) {\mathrm e}^{\frac {i \sqrt {-n^{2}}\, x^{2}+i n^{2}-n \,x^{2}}{x n}}+c_{2} \operatorname {HeunD}\left (-8 \left (-n^{2}\right )^{\frac {1}{4}}, \frac {-8 i \left (-n^{2}\right )^{\frac {3}{4}}-n +8 \sqrt {-n^{2}}\, n}{n}, -\frac {16 i \left (-n^{2}\right )^{\frac {3}{4}}}{n}, \frac {n -8 i \left (-n^{2}\right )^{\frac {3}{4}}-8 \sqrt {-n^{2}}\, n}{n}, \frac {\left (-n^{2}\right )^{\frac {1}{4}} x -i n}{\left (-n^{2}\right )^{\frac {1}{4}} x +i n}\right ) {\mathrm e}^{\frac {-i \sqrt {-n^{2}}\, x^{2}-i n^{2}-n \,x^{2}}{x n}}}{\sqrt {x}} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[x^4*y''[x]+2*x^3*(1+x)*y'[x]+n^2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
Not solved