12.13.13 problem 13

Internal problem ID [1904]
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 : 13
Date solved : Saturday, March 29, 2025 at 11:42:49 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (2 x^{2}+x +1\right ) y^{\prime \prime }+\left (1+7 x \right ) y^{\prime }+2 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 )&=0 \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 18

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

\[ y = 1-\frac {1}{4} \left (x -1\right )^{2}+\frac {13}{48} \left (x -1\right )^{3}-\frac {77}{384} \left (x -1\right )^{4}+\frac {287}{2560} \left (x -1\right )^{5}+\operatorname {O}\left (\left (x -1\right )^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 41

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

\[ y(x)\to \frac {287 (x-1)^5}{2560}-\frac {77}{384} (x-1)^4+\frac {13}{48} (x-1)^3-\frac {1}{4} (x-1)^2+1 \]

✓ Sympy. Time used: 0.884 (sec). Leaf size: 60

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

\[ y{\left (x \right )} = C_{2} \left (x - \frac {73 \left (x - 1\right )^{4}}{192} + \frac {17 \left (x - 1\right )^{3}}{24} - \left (x - 1\right )^{2} - 1\right ) + C_{1} \left (- \frac {77 \left (x - 1\right )^{4}}{384} + \frac {13 \left (x - 1\right )^{3}}{48} - \frac {\left (x - 1\right )^{2}}{4} + 1\right ) + O\left (x^{6}\right ) \]