12.13.49 problem 48

Internal problem ID [1940]
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 : 48
Date solved : Tuesday, March 04, 2025 at 01:46:33 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (-2 x^{2}+x \right ) y^{\prime \prime }+\left (-x^{2}+3 x +1\right ) y^{\prime }+\left (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 )&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 16

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

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

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 32

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

\[ y(x)\to -\frac {1}{8} (x-1)^5+\frac {1}{6} (x-1)^3+\frac {3}{2} (x-1)^2+1 \]

✓ Sympy. Time used: 0.982 (sec). Leaf size: 54

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

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