12.13.44 problem 43

Internal problem ID [1935]
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 : 43
Date solved : Tuesday, March 04, 2025 at 01:46:28 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (1+x \right ) y^{\prime \prime }+\left (2 x^{2}-3 x +1\right ) y^{\prime }-\left (x -4\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 )&=-2\\ y^{\prime }\left (1\right )&=3 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 20

Order:=6; 
ode:=(1+x)*diff(diff(y(x),x),x)+(2*x^2-3*x+1)*diff(y(x),x)-(-4+x)*y(x) = 0; 
ic:=y(1) = -2, D(y)(1) = 3; 
dsolve([ode,ic],y(x),type='series',x=1);
 

\[ y = -2+3 \left (x -1\right )+\frac {3}{2} \left (x -1\right )^{2}-\frac {17}{12} \left (x -1\right )^{3}-\frac {1}{12} \left (x -1\right )^{4}+\frac {1}{8} \left (x -1\right )^{5}+\operatorname {O}\left (\left (x -1\right )^{6}\right ) \]

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

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

\[ y(x)\to -\frac {13}{240} (x-1)^5-\frac {1}{96} (x-1)^4-\frac {2}{3} (x-1)^3+\frac {9}{4} (x-1)^2+3 (x-1)-2 \]

✓ Sympy. Time used: 0.928 (sec). Leaf size: 53

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((4 - x)*y(x) + (x + 1)*Derivative(y(x), (x, 2)) + (2*x**2 - 3*x + 1)*Derivative(y(x), x),0) 
ics = {y(1): -2, Subs(Derivative(y(x), x), x, 1): 3} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=1,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {5 \left (x - 1\right )^{4}}{48} + \frac {5 \left (x - 1\right )^{3}}{24} - \frac {3 \left (x - 1\right )^{2}}{4} + 1\right ) + C_{1} \left (x + \frac {\left (x - 1\right )^{4}}{24} - \frac {\left (x - 1\right )^{3}}{3} - 1\right ) + O\left (x^{6}\right ) \]