31.28 problem 209

Internal problem ID [11032]
Internal file name [OUTPUT/10289_Sunday_December_31_2023_04_04_24_PM_7610360/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 2.1.2-6 Equation of form \((a_3 x^3+a_2 x^2 x+a_1 x+a_0) y''+f(x)y'+g(x)y=0\)
Problem number: 209.
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.

\[ \boxed {\left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime \prime }+\left (\lambda ^{3}+x^{3}\right ) y^{\prime }-\left (\lambda ^{2}-x \lambda +x^{2}\right ) y=0} \]

Maple trace Kovacic algorithm successful

`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 
   A Liouvillian solution exists 
   Reducible group (found an exponential solution) 
   Group is reducible, not completely reducible 
   Solution has integrals. Trying a special function solution free of integrals... 
   -> Trying a solution in terms of special functions: 
      -> Bessel 
      -> elliptic 
      -> Legendre 
      -> Kummer 
         -> 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 
      -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius 
   No special function solution was found. 
<- Kovacics algorithm successful`
 

✓ Solution by Maple

Time used: 0.532 (sec). Leaf size: 76

dsolve((a*x^3+b*x^2+c*x+d)*diff(y(x),x$2)+(x^3+lambda^3)*diff(y(x),x)-(x^2-lambda*x+lambda^2)*y(x)=0,y(x), singsol=all)
 

\[ y \left (x \right ) = \left (x +\lambda \right ) \left (\left (\int {\mathrm e}^{-\left (\int \frac {x^{4}+\left (2 a +\lambda \right ) x^{3}+2 x^{2} b +\left (\lambda ^{3}+2 c \right ) x +\lambda ^{4}+2 d}{\left (a \,x^{3}+x^{2} b +c x +d \right ) \left (x +\lambda \right )}d x \right )}d x \right ) c_{2} +c_{1} \right ) \]

✓ Solution by Mathematica

Time used: 1.343 (sec). Leaf size: 240

DSolve[(a*x^3+b*x^2+c*x+d)*y''[x]+(x^3+\[Lambda]^3)*y'[x]-(x^2-\[Lambda]*x+\[Lambda]^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {c_2 (\lambda +x) \int _1^x\exp \left (-\frac {\lambda +K[1]+2 a \log (\lambda +K[1])+\text {RootSum}\left [-a \lambda ^3+b \lambda ^2+3 a \text {$\#$1} \lambda ^2-3 a \text {$\#$1}^2 \lambda -c \lambda -2 b \text {$\#$1} \lambda +a \text {$\#$1}^3+b \text {$\#$1}^2+d+c \text {$\#$1}\&,\frac {a \log (\lambda +K[1]-\text {$\#$1}) \lambda ^3-b \log (\lambda +K[1]-\text {$\#$1}) \lambda ^2+c \log (\lambda +K[1]-\text {$\#$1}) \lambda +2 b \log (\lambda +K[1]-\text {$\#$1}) \text {$\#$1} \lambda -b \log (\lambda +K[1]-\text {$\#$1}) \text {$\#$1}^2-d \log (\lambda +K[1]-\text {$\#$1})-c \log (\lambda +K[1]-\text {$\#$1}) \text {$\#$1}}{3 a \lambda ^2-2 b \lambda -6 a \text {$\#$1} \lambda +3 a \text {$\#$1}^2+c+2 b \text {$\#$1}}\&\right ]}{a}\right )dK[1]}{\lambda }+\frac {c_1 (\lambda +x)}{\lambda } \]