12.13.17 problem 20

Internal problem ID [1908]
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 : 20
Date solved : Saturday, March 29, 2025 at 11:42:55 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (1+2 x \right ) y^{\prime \prime }-\left (1-2 x \right ) y^{\prime }-\left (3-2 x \right ) 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: 0.005 (sec). Leaf size: 20

Order:=6; 
ode:=(2*x+1)*diff(diff(y(x),x),x)-(1-2*x)*diff(y(x),x)-(3-2*x)*y(x) = 0; 
ic:=y(1) = 1, D(y)(1) = -2; 
dsolve([ode,ic],y(x),type='series',x=1);
 

\[ y = 1-2 \left (-1+x \right )+\frac {1}{2} \left (-1+x \right )^{2}-\frac {1}{6} \left (-1+x \right )^{3}+\frac {5}{36} \left (-1+x \right )^{4}-\frac {73}{1080} \left (-1+x \right )^{5}+\operatorname {O}\left (\left (-1+x \right )^{6}\right ) \]

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

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

\[ y(x)\to -\frac {73 (x-1)^5}{1080}+\frac {5}{36} (x-1)^4-\frac {1}{6} (x-1)^3+\frac {1}{2} (x-1)^2-2 (x-1)+1 \]

✓ Sympy. Time used: 0.873 (sec). Leaf size: 48

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-(3 - 2*x)*y(x) + (2*x - 1)*Derivative(y(x), x) + (2*x + 1)*Derivative(y(x), (x, 2)),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)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {\left (x - 1\right )^{4}}{18} - \frac {\left (x - 1\right )^{3}}{6} + \frac {\left (x - 1\right )^{2}}{6} + 1\right ) + C_{1} \left (x - \frac {\left (x - 1\right )^{4}}{24} - \frac {\left (x - 1\right )^{2}}{6} - 1\right ) + O\left (x^{6}\right ) \]