55.28.46 problem 56

Internal problem ID [13829]
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 : 56
Date solved : Thursday, October 02, 2025 at 08:07:23 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Maple. Time used: 0.041 (sec). Leaf size: 72

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

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

✓ Mathematica. Time used: 60.052 (sec). Leaf size: 74

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

\begin{align*} y(x)&\to e^{x \left (-\frac {a x^n}{n+1}-\frac {b x^m}{m+1}\right )} \left (\int _1^x\exp \left (K[1] \left (\frac {b K[1]^m}{m+1}+\frac {a K[1]^n}{n+1}\right )\right ) c_1dK[1]+c_2\right ) \end{align*}

✗ 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*n*x**(n - 1) + b*m*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