69.24.2 problem 740

Internal problem ID [18501]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 18.2. Expanding a solution in generalized power series. Bessels equation. Exercises page 177
Problem number : 740
Date solved : Thursday, October 02, 2025 at 03:14:24 PM
CAS classification : [_separable]

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

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 83

Order:=6; 
ode:=(1+x)*diff(y(x),x)-n*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1+n x +\frac {n \left (n -1\right ) x^{2}}{2}+\frac {n \left (n^{2}-3 n +2\right ) x^{3}}{6}+\frac {n \left (n^{3}-6 n^{2}+11 n -6\right ) x^{4}}{24}+\frac {n \left (n^{4}-10 n^{3}+35 n^{2}-50 n +24\right ) x^{5}}{120}\right ) y \left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 143

ode=(1+x)*D[y[x],x]-n*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {n^5 x^5}{120}-\frac {n^4 x^5}{12}+\frac {n^4 x^4}{24}+\frac {7 n^3 x^5}{24}-\frac {n^3 x^4}{4}+\frac {n^3 x^3}{6}-\frac {5 n^2 x^5}{12}+\frac {11 n^2 x^4}{24}-\frac {n^2 x^3}{2}+\frac {n^2 x^2}{2}+\frac {n x^5}{5}-\frac {n x^4}{4}+\frac {n x^3}{3}-\frac {n x^2}{2}+n x+1\right ) \]

✓ Sympy. Time used: 0.379 (sec). Leaf size: 134

from sympy import * 
x = symbols("x") 
n = symbols("n") 
y = Function("y") 
ode = Eq(-n*y(x) + (x + 1)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="1st_power_series",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} + C_{1} n x + \frac {C_{1} n x^{2} \left (n - 1\right )}{2} + \frac {C_{1} n x^{3} \left (n \left (n - 1\right ) - 2 n + 2\right )}{6} + \frac {C_{1} n x^{4} \left (- 3 n \left (n - 1\right ) + n \left (n \left (n - 1\right ) - 2 n + 2\right ) + 6 n - 6\right )}{24} + \frac {C_{1} n x^{5} \left (12 n \left (n - 1\right ) - 4 n \left (n \left (n - 1\right ) - 2 n + 2\right ) + n \left (- 3 n \left (n - 1\right ) + n \left (n \left (n - 1\right ) - 2 n + 2\right ) + 6 n - 6\right ) - 24 n + 24\right )}{120} + O\left (x^{6}\right ) \]