61.27.45 problem 55

Internal problem ID [12476]
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
Problem number : 55
Date solved : Monday, March 31, 2025 at 05:35:54 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+\left (a \,x^{n}+b \,x^{m}\right ) y^{\prime }-\left (a \,x^{n -1}+b \,x^{m -1}\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.245 (sec). Leaf size: 47

ode:=diff(diff(y(x),x),x)+(a*x^n+b*x^m)*diff(y(x),x)-(a*x^(n-1)+b*x^(m-1))*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = x \left (\int \frac {{\mathrm e}^{-\frac {\left (b \left (n +1\right ) x^{m}+a \,x^{n} \left (m +1\right )\right ) x}{\left (n +1\right ) \left (m +1\right )}}}{x^{2}}d x c_2 +c_1 \right ) \]

✓ Mathematica. Time used: 0.818 (sec). Leaf size: 55

ode=D[y[x],{x,2}]+(a*x^n+b*x^m)*D[y[x],x]-(a*x^(n-1)+b*x^(m-1))*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to x \left (c_2 \int _1^x\frac {\exp \left (K[1] \left (-\frac {b K[1]^m}{m+1}-\frac {a K[1]^n}{n+1}\right )\right )}{K[1]^2}dK[1]+c_1\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
m = symbols("m") 
n = symbols("n") 
y = Function("y") 
ode = Eq((a*x**n + b*x**m)*Derivative(y(x), x) - (a*x**(n - 1) + b*x**(m - 1))*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

TypeError : Add object cannot be interpreted as an integer