9.8.19 problem problem 19

Internal problem ID [1084]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Chapter 11 Power series methods. Section 11.2 Power series solutions. Page 624
Problem number : problem 19
Date solved : Saturday, March 29, 2025 at 10:38:30 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} \left (-x^{2}+2 x \right ) y^{\prime \prime }-6 \left (x -1\right ) y^{\prime }-4 y&=0 \end{align*}

Using series method with expansion around

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

With initial conditions

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 14

Order:=6; 
ode:=(-x^2+2*x)*diff(diff(y(x),x),x)-6*(x-1)*diff(y(x),x)-4*y(x) = 0; 
ic:=y(1) = 0, D(y)(1) = 1; 
dsolve([ode,ic],y(x),type='series',x=1);
 

\[ y = \left (x -1\right )+\frac {5}{3} \left (x -1\right )^{3}+\frac {7}{3} \left (x -1\right )^{5}+\operatorname {O}\left (\left (x -1\right )^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 24

ode=(2*x-x^2)*D[y[x],{x,2}]-6*(x-1)*D[y[x],x]-4*y[x]==0; 
ic={y[1]==0,Derivative[1][y][1]==1}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,1,5}]
 

\[ y(x)\to \frac {7}{3} (x-1)^5+\frac {5}{3} (x-1)^3+x-1 \]

✓ Sympy. Time used: 0.873 (sec). Leaf size: 36

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((6 - 6*x)*Derivative(y(x), x) + (-x**2 + 2*x)*Derivative(y(x), (x, 2)) - 4*y(x),0) 
ics = {y(1): 0, Subs(Derivative(y(x), x), x, 1): 1} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=1,n=6)
 

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