40.17.8 problem 18

Internal problem ID [6809]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 26. Integration in series (singular points). Supplemetary problems. Page 218
Problem number : 18
Date solved : Sunday, March 30, 2025 at 11:23:23 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (x +1\right ) y^{\prime \prime }+x \left (x +1\right ) y^{\prime }-y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.021 (sec). Leaf size: 37

Order:=6; 
ode:=x^2*(1+x)*diff(diff(y(x),x),x)+x*(1+x)*diff(y(x),x)-y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 x \left (1-\frac {1}{3} x +\frac {1}{6} x^{2}-\frac {1}{10} x^{3}+\frac {1}{15} x^{4}-\frac {1}{21} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (-2-2 x +\operatorname {O}\left (x^{6}\right )\right )}{x} \]

✓ Mathematica. Time used: 0.027 (sec). Leaf size: 45

ode=x^2*(x+1)*D[y[x],{x,2}]+x*(x+1)*D[y[x],x]-y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (\frac {x^5}{15}-\frac {x^4}{10}+\frac {x^3}{6}-\frac {x^2}{3}+x\right )+c_1 \left (\frac {1}{x}+1\right ) \]

✓ Sympy. Time used: 1.023 (sec). Leaf size: 12

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(x + 1)*Derivative(y(x), (x, 2)) + x*(x + 1)*Derivative(y(x), x) - y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} x + \frac {C_{1}}{x} + O\left (x^{6}\right ) \]