52.4.3 problem 11

Internal problem ID [8313]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 6 SERIES SOLUTIONS OF LINEAR EQUATIONS. CHAPTER 6 IN REVIEW. Page 271
Problem number : 11
Date solved : Wednesday, March 05, 2025 at 05:34:51 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} \left (x -1\right ) y^{\prime \prime }+3 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 74

Order:=8; 
ode:=(x-1)*diff(diff(y(x),x),x)+3*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1+\frac {3}{2} x^{2}+\frac {1}{2} x^{3}+\frac {5}{8} x^{4}+\frac {9}{20} x^{5}+\frac {29}{80} x^{6}+\frac {163}{560} x^{7}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{3}+\frac {1}{4} x^{4}+\frac {9}{40} x^{5}+\frac {7}{40} x^{6}+\frac {79}{560} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 91

ode=(x-1)*D[y[x],{x,2}]+3*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_2 \left (\frac {79 x^7}{560}+\frac {7 x^6}{40}+\frac {9 x^5}{40}+\frac {x^4}{4}+\frac {x^3}{2}+x\right )+c_1 \left (\frac {163 x^7}{560}+\frac {29 x^6}{80}+\frac {9 x^5}{20}+\frac {5 x^4}{8}+\frac {x^3}{2}+\frac {3 x^2}{2}+1\right ) \]

✓ Sympy. Time used: 0.916 (sec). Leaf size: 70

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((x - 1)*Derivative(y(x), (x, 2)) + 3*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {29 x^{6}}{80} + \frac {9 x^{5}}{20} + \frac {5 x^{4}}{8} + \frac {x^{3}}{2} + \frac {3 x^{2}}{2} + 1\right ) + C_{1} x \left (\frac {7 x^{5}}{40} + \frac {9 x^{4}}{40} + \frac {x^{3}}{4} + \frac {x^{2}}{2} + 1\right ) + O\left (x^{8}\right ) \]