6.13.23 problem 26

Internal problem ID [1914]
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 : 26
Date solved : Tuesday, September 30, 2025 at 05:21:13 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (10-2 x \right ) y^{\prime \prime }+\left (1+x \right ) y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 2 \end{align*}

With initial conditions

\begin{align*} y \left (2\right )&=2 \\ y^{\prime }\left (2\right )&=-4 \\ \end{align*}

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

Order:=6; 
ode:=(10-2*x)*diff(diff(y(x),x),x)+(1+x)*y(x) = 0; 
ic:=[y(2) = 2, D(y)(2) = -4]; 
dsolve([ode,op(ic)],y(x),type='series',x=2);
 

\[ y = 2-4 \left (x -2\right )-\frac {1}{2} \left (x -2\right )^{2}+\frac {2}{9} \left (x -2\right )^{3}+\frac {49}{432} \left (x -2\right )^{4}+\frac {23}{1080} \left (x -2\right )^{5}+\operatorname {O}\left (\left (x -2\right )^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 46

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

\[ y(x)\to \frac {23 (x-2)^5}{1080}+\frac {49}{432} (x-2)^4+\frac {2}{9} (x-2)^3-\frac {1}{2} (x-2)^2-4 (x-2)+2 \]

✓ Sympy. Time used: 0.280 (sec). Leaf size: 48

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((10 - 2*x)*Derivative(y(x), (x, 2)) + (x + 1)*y(x),0) 
ics = {y(2): 2, Subs(Derivative(y(x), x), x, 2): -4} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=2,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {\left (x - 2\right )^{4}}{864} - \frac {\left (x - 2\right )^{3}}{18} - \frac {\left (x - 2\right )^{2}}{4} + 1\right ) + C_{1} \left (x - \frac {\left (x - 2\right )^{4}}{36} - \frac {\left (x - 2\right )^{3}}{12} - 2\right ) + O\left (x^{6}\right ) \]