46.1.21 problem 19

Internal problem ID [9524]
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. Section 6.2 SOLUTIONS ABOUT ORDINARY POINTS. EXERCISES 6.2. Page 246
Problem number : 19
Date solved : Tuesday, September 30, 2025 at 06:20:03 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

With initial conditions

\begin{align*} y \left (0\right )&=-2 \\ y^{\prime }\left (0\right )&=6 \\ \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 24

Order:=8; 
ode:=(x-1)*diff(diff(y(x),x),x)-x*diff(y(x),x)+y(x) = 0; 
ic:=[y(0) = -2, D(y)(0) = 6]; 
dsolve([ode,op(ic)],y(x),type='series',x=0);
 

\[ y = -2+6 x -x^{2}-\frac {1}{3} x^{3}-\frac {1}{12} x^{4}-\frac {1}{60} x^{5}-\frac {1}{360} x^{6}-\frac {1}{2520} x^{7}+\operatorname {O}\left (x^{8}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 48

ode=(x-1)*D[y[x],{x,2}]-x*D[y[x],x]+y[x]==0; 
ic={y[0]==-2,Derivative[1][y][0] ==6}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to -\frac {x^7}{2520}-\frac {x^6}{360}-\frac {x^5}{60}-\frac {x^4}{12}-\frac {x^3}{3}-x^2+6 x-2 \]

✓ Sympy. Time used: 0.227 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*Derivative(y(x), x) + (x - 1)*Derivative(y(x), (x, 2)) + y(x),0) 
ics = {y(0): -2, Subs(Derivative(y(x), x), x, 0): 6} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{6}}{720} + \frac {x^{5}}{120} + \frac {x^{4}}{24} + \frac {x^{3}}{6} + \frac {x^{2}}{2} + 1\right ) + C_{1} x + O\left (x^{8}\right ) \]