Internal problem ID [10872]
Internal file name [OUTPUT/10129_Sunday_December_24_2023_05_13_46_PM_18930705/index.tex]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-2 Equation of form
\(y''+f(x)y'+g(x)y=0\)
Problem number: 49.
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.
\[ \boxed {y^{\prime \prime }+2 a \,x^{n} y^{\prime }+\left (x^{2 n} a^{2}+b \,x^{2 m}+x^{n -1} a n +c \,x^{m -1}\right ) y=0} \]
Maple trace
`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 <- No Liouvillian solutions exists -> Trying a solution in terms of special functions: -> Bessel -> elliptic -> Legendre -> Kummer -> hyper3: Equivalence to 1F1 under a power @ Moebius <- hyper3 successful: received ODE is equivalent to the 1F1 ODE <- Kummer successful <- special function solution successful`
✓ Solution by Maple
Time used: 0.406 (sec). Leaf size: 147
dsolve(diff(y(x),x$2)+2*a*x^n*diff(y(x),x)+(a^2*x^(2*n)+b*x^(2*m)+a*n*x^(n-1)+c*x^(m-1))*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = x \left (\operatorname {KummerM}\left (\frac {\left (m +2\right ) \sqrt {b}+i c}{\sqrt {b}\, \left (2 m +2\right )}, \frac {m +2}{1+m}, \frac {2 i \sqrt {b}\, x^{1+m}}{1+m}\right ) c_{1} +\operatorname {KummerU}\left (\frac {\left (m +2\right ) \sqrt {b}+i c}{\sqrt {b}\, \left (2 m +2\right )}, \frac {m +2}{1+m}, \frac {2 i \sqrt {b}\, x^{1+m}}{1+m}\right ) c_{2} \right ) {\mathrm e}^{\frac {-i \left (n +1\right ) \sqrt {b}\, x^{1+m}-x^{n +1} a \left (1+m \right )}{\left (n +1\right ) \left (1+m \right )}} \]
✓ Solution by Mathematica
Time used: 0.376 (sec). Leaf size: 236
DSolve[y''[x]+2*a*x^n*y'[x]+(a^2*x^(2*n)+b*x^(2*m)+a*n*x^(n-1)+c*x^(m-1))*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to 2^{\frac {m}{2 m+2}} x^{-m/2} \left (x^{m+1}\right )^{\frac {m}{2 m+2}} \exp \left (-x \left (\frac {a x^n}{n+1}+\frac {\sqrt {b} x^m}{\sqrt {-(m+1)^2}}\right )\right ) \left (c_1 \operatorname {HypergeometricU}\left (-\frac {(m+1) \left (m c+c+\sqrt {b} m \sqrt {-(m+1)^2}\right )}{2 \sqrt {b} \left (-(m+1)^2\right )^{3/2}},\frac {m}{m+1},\frac {2 \sqrt {b} x^{m+1}}{\sqrt {-(m+1)^2}}\right )+c_2 L_{\frac {(m+1) \left (m c+c+\sqrt {b} m \sqrt {-(m+1)^2}\right )}{2 \sqrt {b} \left (-(m+1)^2\right )^{3/2}}}^{-\frac {1}{m+1}}\left (\frac {2 \sqrt {b} x^{m+1}}{\sqrt {-(m+1)^2}}\right )\right ) \]