12.13.22 problem 25

Internal problem ID [1913]
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 : 25
Date solved : Tuesday, March 04, 2025 at 01:46:06 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

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

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

\[ y = -2+3 \left (x +2\right )-\frac {9}{2} \left (x +2\right )^{2}+\frac {11}{6} \left (x +2\right )^{3}+\frac {5}{24} \left (x +2\right )^{4}+\frac {7}{20} \left (x +2\right )^{5}+\operatorname {O}\left (\left (x +2\right )^{6}\right ) \]

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

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

\[ y(x)\to \frac {7}{20} (x+2)^5+\frac {5}{24} (x+2)^4+\frac {11}{6} (x+2)^3-\frac {9}{2} (x+2)^2+3 (x+2)-2 \]

✓ Sympy. Time used: 0.835 (sec). Leaf size: 49

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

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