84.34.7 problem 20.20

Internal problem ID [22334]
Book : Schaums outline series. Differential Equations By Richard Bronson. 1973. McGraw-Hill Inc. ISBN 0-07-008009-7
Section : Chapter 20. Regular singular points and the method of Frobenius. Supplementary problems
Problem number : 20.20
Date solved : Thursday, October 02, 2025 at 08:37:40 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 58

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

\[ y = c_1 \,x^{2} \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\ln \left (x \right ) \left (2 x^{2}+2 x^{3}+x^{4}+\frac {1}{3} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2+2 x +3 x^{2}+x^{3}-\frac {1}{9} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 73

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

\[ y(x)\to c_1 \left (\frac {1}{4} \left (x^4-4 x^2-4 x+4\right )-\frac {1}{2} x^2 \left (x^2+2 x+2\right ) \log (x)\right )+c_2 \left (\frac {x^6}{24}+\frac {x^5}{6}+\frac {x^4}{2}+x^3+x^2\right ) \]

✓ Sympy. Time used: 0.417 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), (x, 2)) - (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)
 

\[ y{\left (x \right )} = C_{1} x^{2} \left (\frac {x^{3}}{6} + \frac {x^{2}}{2} + x + 1\right ) + O\left (x^{6}\right ) \]