6.12.23 problem 25

Internal problem ID [1877]
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 : 25
Date solved : Tuesday, September 30, 2025 at 05:20:49 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}-8 x +14\right ) y^{\prime \prime }-8 \left (x -4\right ) y^{\prime }+20 y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 4 \end{align*}

With initial conditions

\begin{align*} y \left (4\right )&=3 \\ y^{\prime }\left (4\right )&=-4 \\ \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 20

Order:=6; 
ode:=(x^2-8*x+14)*diff(diff(y(x),x),x)-8*(-4+x)*diff(y(x),x)+20*y(x) = 0; 
ic:=[y(4) = 3, D(y)(4) = -4]; 
dsolve([ode,op(ic)],y(x),type='series',x=4);
 

\[ y = 3-4 \left (x -4\right )+15 \left (x -4\right )^{2}-4 \left (x -4\right )^{3}+\frac {15}{4} \left (x -4\right )^{4}-\frac {1}{5} \left (x -4\right )^{5}+\operatorname {O}\left (\left (x -4\right )^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 44

ode=(x^2-8*x+14)*D[y[x],{x,2}]+8*(x-4)*D[y[x],x]+20*y[x]==0; 
ic={y[4]==3,Derivative[1][y][4]==-4}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,4,5}]
 

\[ y(x)\to -\frac {35}{3} (x-4)^5+\frac {95}{4} (x-4)^4-\frac {28}{3} (x-4)^3+15 (x-4)^2-4 (x-4)+3 \]

✓ Sympy. Time used: 0.272 (sec). Leaf size: 34

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((32 - 8*x)*Derivative(y(x), x) + (x**2 - 8*x + 14)*Derivative(y(x), (x, 2)) + 20*y(x),0) 
ics = {y(4): 3, Subs(Derivative(y(x), x), x, 4): -4} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=4,n=6)
 

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