12.12.20 problem 22

Internal problem ID [1874]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.2 SERIES SOLUTIONS NEAR AN ORDINARY POINT I. Exercises 7.2. Page 329
Problem number : 22
Date solved : Saturday, March 29, 2025 at 11:42:05 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 20

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

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

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 42

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

\[ y(x)\to \frac {3}{5} (x-3)^5-\frac {5}{4} (x-3)^4-2 (x-3)^3+3 (x-3)^2+3 (x-3)-2 \]

✓ Sympy. Time used: 0.744 (sec). Leaf size: 39

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

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