31.26 problem 207

Internal problem ID [11030]
Internal file name [OUTPUT/10287_Sunday_December_31_2023_04_01_34_PM_76722025/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: 207.
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.

\[ \boxed {2 \left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime \prime }+3 \left (3 a \,x^{2}+2 b x +c \right ) y^{\prime }+\left (6 x a +2 b +\lambda \right ) y=0} \]

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 
   Group is reducible or imprimitive 
   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: 0.531 (sec). Leaf size: 101

dsolve(2*(a*x^3+b*x^2+c*x+d)*diff(y(x),x$2)+3*(3*a*x^2+2*b*x+c)*diff(y(x),x)+(6*a*x+2*b+lambda)*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 135.727 (sec). Leaf size: 3202

DSolve[2*(a*x^3+b*x^2+c*x+d)*y''[x]+3*(3*a*x^2+2*b*x+c)*y'[x]+(6*a*x+2*b+\[Lambda])*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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