2.2.12 Problem 12

Internal problem ID [8816]
Book : Own collection of miscellaneous problems
Section : section 2.0
Problem number : 12
Date solved : Thursday, March 13, 2025 at 06:09:46 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

Solve

$$\begin{array}{rl}{y}^{\prime \prime }-ax{y}^{\prime }-bxy-c{x}^2& =0\end{array}$$

✓ Maple. Time used: 0.024 (sec). Leaf size: 95

ode:=diff(diff(y(x),x),x)-a*x*diff(y(x),x)-b*x*y(x)-c*x^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

$$y=\frac{{\mathrm{e}}^{-\frac{bx}{a}}\mathrm{KummerM}(-\frac{b^2}{2a^3},\frac{1}{2},\frac{{(a^{2}x+2b)}^2}{2a^3})c_2{b}^2+{\mathrm{e}}^{-\frac{bx}{a}}\mathrm{KummerU}(-\frac{b^2}{2a^3},\frac{1}{2},\frac{{(a^{2}x+2b)}^2}{2a^3})c_1{b}^2+c(-bx+a)}{b^2}$$

Maple trace

`Methods for second order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying high order exact linear fully integrable 
trying differential order: 2; linear nonhomogeneous with symmetry [0,1] 
trying a double symmetry of the form [xi=0, eta=F(x)] 
-> Try solving first the homogeneous part of the ODE 
   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 
      -> Kummer 
         -> hyper3: Equivalence to 1F1 under a power @ Moebius 
      -> hypergeometric 
         -> heuristic approach 
         <- heuristic approach successful 
      <- hypergeometric successful 
   <- special function solution successful 
<- solving first the homogeneous part of the ODE successful`
 

✓ Mathematica. Time used: 2.789 (sec). Leaf size: 569

ode=D[y[x],{x,2}]-a*x*D[y[x],x]-b*x*y[x]-c*x^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

$$y(x)\to e^{-\frac{bx}{a}}(\mathrm{HermiteH}(\frac{b^2}{a^3},\frac{x{a}^2+2b}{\sqrt{2}{a}^{3/2}}){\int }_1^x\frac{a^{4}c{e}^{\frac{bK[1]}{a}}\mathrm{Hypergeometric}1\mathrm{F}1(-\frac{b^2}{2a^3},\frac{1}{2},\frac{{(K[1]a^2+2b)}^2}{2a^3})K[1{]}^2}{b^2(\sqrt{2}\mathrm{HermiteH}(\frac{b^2}{a^3}-1,\frac{K[1]a^2+2b}{\sqrt{2}{a}^{3/2}})\mathrm{Hypergeometric}1\mathrm{F}1(-\frac{b^2}{2a^3},\frac{1}{2},\frac{{(K[1]a^2+2b)}^2}{2a^3})a^{3/2}+\mathrm{HermiteH}(\frac{b^2}{a^3},\frac{K[1]a^2+2b}{\sqrt{2}{a}^{3/2}})\mathrm{Hypergeometric}1\mathrm{F}1(1-\frac{b^2}{2a^3},\frac{3}{2},\frac{{(K[1]a^2+2b)}^2}{2a^3})(K[1]a^2+2b))}dK[1]+\mathrm{Hypergeometric}1\mathrm{F}1(-\frac{b^2}{2a^3},\frac{1}{2},\frac{{(x{a}^2+2b)}^2}{2a^3}){\int }_1^x-\frac{a^{4}c{e}^{\frac{bK[2]}{a}}\mathrm{HermiteH}(\frac{b^2}{a^3},\frac{K[2]a^2+2b}{\sqrt{2}{a}^{3/2}})K[2{]}^2}{b^2(\sqrt{2}\mathrm{HermiteH}(\frac{b^2}{a^3}-1,\frac{K[2]a^2+2b}{\sqrt{2}{a}^{3/2}})\mathrm{Hypergeometric}1\mathrm{F}1(-\frac{b^2}{2a^3},\frac{1}{2},\frac{{(K[2]a^2+2b)}^2}{2a^3})a^{3/2}+\mathrm{HermiteH}(\frac{b^2}{a^3},\frac{K[2]a^2+2b}{\sqrt{2}{a}^{3/2}})\mathrm{Hypergeometric}1\mathrm{F}1(1-\frac{b^2}{2a^3},\frac{3}{2},\frac{{(K[2]a^2+2b)}^2}{2a^3})(K[2]a^2+2b))}dK[2]+c_2\mathrm{Hypergeometric}1\mathrm{F}1(-\frac{b^2}{2a^3},\frac{1}{2},\frac{{(x{a}^2+2b)}^2}{2a^3})+c_1\mathrm{HermiteH}(\frac{b^2}{a^3},\frac{x{a}^2+2b}{\sqrt{2}{a}^{3/2}}))$$

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
y = Function("y") 
ode = Eq(-a*x*Derivative(y(x), x) - b*x*y(x) - c*x**2 + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-x*(b*y(x) + c*x) + Derivative(y(x), (x, 2)))/(a*x) cannot be solved by the factorable group method