7.14.20 problem 20

Internal problem ID [445]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.2 (Series solution near ordinary points). Problems at page 216
Problem number : 20
Date solved : Tuesday, March 04, 2025 at 11:24:23 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}-6 x +10\right ) y^{\prime \prime }-4 \left (x -3\right ) y^{\prime }+6 y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 3 \end{align*}

With initial conditions

\begin{align*} y \left (3\right )&=2\\ y^{\prime }\left (3\right )&=0 \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 14

Order:=6; 
ode:=(x^2-6*x+10)*diff(diff(y(x),x),x)-4*(x-3)*diff(y(x),x)+6*y(x) = 0; 
ic:=y(3) = 2, D(y)(3) = 0; 
dsolve([ode,ic],y(x),type='series',x=3);
 

\[ y = -6 x^{2}+36 x -52 \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 12

ode=(x^2-6*x+10)*D[y[x],{x,2}]-4*(x-3)*D[y[x],x]+6*y[x]==0; 
ic={y[3]==2,Derivative[1][y][3] ==0}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,3,5}]
 

\[ y(x)\to 2-6 (x-3)^2 \]

✓ Sympy. Time used: 0.797 (sec). Leaf size: 27

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((12 - 4*x)*Derivative(y(x), x) + (x**2 - 6*x + 10)*Derivative(y(x), (x, 2)) + 6*y(x),0) 
ics = {y(3): 2, Subs(Derivative(y(x), x), x, 3): 0} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=3,n=6)
 

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