49.15.6 problem 2

Internal problem ID [7692]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 3. Linear equations with variable coefficients. Page 130
Problem number : 2
Date solved : Sunday, March 30, 2025 at 12:19:03 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

\begin{align*} 1 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 12

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

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 14

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

\[ y(x)\to \frac {1}{6} (x-1)^3+1 \]

✓ Sympy. Time used: 0.799 (sec). Leaf size: 20

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

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