12.13.45 problem 44

Internal problem ID [1936]
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 : 44
Date solved : Tuesday, March 04, 2025 at 01:46:29 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+\left (3 x^{2}+12 x +13\right ) y^{\prime }+\left (5+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: 0.001 (sec). Leaf size: 20

Order:=6; 
ode:=diff(diff(y(x),x),x)+(3*x^2+12*x+13)*diff(y(x),x)+(5+2*x)*y(x) = 0; 
ic:=y(-2) = 2, D(y)(-2) = -3; 
dsolve([ode,ic],y(x),type='series',x=-2);
 

\[ y = 2-3 \left (x +2\right )+\frac {1}{2} \left (x +2\right )^{2}-\frac {1}{3} \left (x +2\right )^{3}+\frac {31}{24} \left (x +2\right )^{4}-\frac {53}{120} \left (x +2\right )^{5}+\operatorname {O}\left (\left (x +2\right )^{6}\right ) \]

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

ode=D[y[x],{x,2}]+(13+12*x+3*x^2)*D[y[x],x]+(5+2*x)*y[x]==0; 
ic={y[-2]==2,Derivative[1][y][-2]==-3}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,-2,5}]
 

\[ y(x)\to -\frac {53}{120} (x+2)^5+\frac {31}{24} (x+2)^4-\frac {1}{3} (x+2)^3+\frac {1}{2} (x+2)^2-3 (x+2)+2 \]

✓ Sympy. Time used: 0.838 (sec). Leaf size: 49

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((2*x + 5)*y(x) + (3*x**2 + 12*x + 13)*Derivative(y(x), x) + Derivative(y(x), (x, 2)),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)
 

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