28.5.17 problem 9.17

Internal problem ID [4604]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 9. Series Solutions of Differential Equations. Problems at page 426
Problem number : 9.17
Date solved : Tuesday, March 04, 2025 at 06:56:23 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }-2 x y^{\prime }-y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.020 (sec). Leaf size: 60

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

\[ y \left (x \right ) = c_{1} x \left (1+\frac {3}{2} x +\frac {5}{4} x^{2}+\frac {35}{48} x^{3}+\frac {21}{64} x^{4}+\frac {77}{640} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (x +\frac {3}{2} x^{2}+\frac {5}{4} x^{3}+\frac {35}{48} x^{4}+\frac {21}{64} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (1+2 x +\frac {7}{4} x^{2}+\frac {11}{12} x^{3}+\frac {61}{192} x^{4}+\frac {131}{1920} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

✓ Mathematica. Time used: 0.026 (sec). Leaf size: 87

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

\[ y(x)\to c_1 \left (\frac {1}{48} x \left (35 x^3+60 x^2+72 x+48\right ) \log (x)+\frac {1}{192} \left (-79 x^4-64 x^3+48 x^2+192 x+192\right )\right )+c_2 \left (\frac {21 x^5}{64}+\frac {35 x^4}{48}+\frac {5 x^3}{4}+\frac {3 x^2}{2}+x\right ) \]

✓ Sympy. Time used: 0.737 (sec). Leaf size: 36

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

\[ y{\left (x \right )} = C_{1} x \left (\frac {21 x^{4}}{64} + \frac {35 x^{3}}{48} + \frac {5 x^{2}}{4} + \frac {3 x}{2} + 1\right ) + O\left (x^{6}\right ) \]