12.13.12 problem 12

Internal problem ID [1903]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.3 SERIES SOLUTIONS NEAR AN ORDINARY POINT II. Exercises 7.3. Page 338
Problem number : 12
Date solved : Tuesday, March 04, 2025 at 01:45:56 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-\left (6-7 x \right ) y^{\prime }+8 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 )&=-2 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 20

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)-(6-7*x)*diff(y(x),x)+8*y(x) = 0; 
ic:=y(1) = 1, D(y)(1) = -2; 
dsolve([ode,ic],y(x),type='series',x=1);
 

\[ y = 1-2 \left (x -1\right )-3 \left (x -1\right )^{2}+8 \left (x -1\right )^{3}-4 \left (x -1\right )^{4}-\frac {42}{5} \left (x -1\right )^{5}+\operatorname {O}\left (\left (x -1\right )^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 40

ode=x^2*D[y[x],{x,2}]-(6-7*x)*D[y[x],x]+8*y[x]==0; 
ic={y[1]==1,Derivative[1][y][1]==-2}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,1,5}]
 

\[ y(x)\to -\frac {42}{5} (x-1)^5-4 (x-1)^4+8 (x-1)^3-3 (x-1)^2-2 (x-1)+1 \]

✓ Sympy. Time used: 0.823 (sec). Leaf size: 56

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) - (6 - 7*x)*Derivative(y(x), x) + 8*y(x),0) 
ics = {y(1): 1, Subs(Derivative(y(x), x), x, 1): -2} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=1,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (x + \frac {7 \left (x - 1\right )^{4}}{2} - 2 \left (x - 1\right )^{3} - \frac {\left (x - 1\right )^{2}}{2} - 1\right ) + C_{1} \left (3 \left (x - 1\right )^{4} + 4 \left (x - 1\right )^{3} - 4 \left (x - 1\right )^{2} + 1\right ) + O\left (x^{6}\right ) \]