87.25.19 problem 19

Internal problem ID [23822]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 5. Series solutions of second order linear equations. Exercise at page 232
Problem number : 19
Date solved : Thursday, October 02, 2025 at 09:45:31 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-2 \left (x -1\right ) y^{\prime }+2 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: 0.001 (sec). Leaf size: 14

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

\[ y = 1-\left (x -1\right )^{2}-\frac {1}{6} \left (x -1\right )^{4}+\operatorname {O}\left (\left (x -1\right )^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 21

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

\[ y(x)\to -\frac {1}{6} (x-1)^4-(x-1)^2+1 \]

✓ Sympy. Time used: 0.201 (sec). Leaf size: 26

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

\[ y{\left (x \right )} = C_{2} \left (- \frac {\left (x - 1\right )^{4}}{6} - \left (x - 1\right )^{2} + 1\right ) + C_{1} \left (x - 1\right ) + O\left (x^{6}\right ) \]