6.13.47 problem 46

Internal problem ID [1938]
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 : 46
Date solved : Tuesday, September 30, 2025 at 05:21:30 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}+4 x +3\right ) y^{\prime \prime }-\left (-x^{2}+4 x +5\right ) 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 )&=-1 \\ \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 20

Order:=6; 
ode:=(x^2+4*x+3)*diff(diff(y(x),x),x)-(-x^2+4*x+5)*diff(y(x),x)-(x+2)*y(x) = 0; 
ic:=[y(-2) = 2, D(y)(-2) = -1]; 
dsolve([ode,op(ic)],y(x),type='series',x=-2);
 

\[ y = 2-\left (2+x \right )-\frac {7}{2} \left (2+x \right )^{2}-\frac {43}{6} \left (2+x \right )^{3}-\frac {203}{24} \left (2+x \right )^{4}-\frac {167}{30} \left (2+x \right )^{5}+\operatorname {O}\left (\left (2+x \right )^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 43

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

\[ y(x)\to -\frac {167}{30} (x+2)^5-\frac {203}{24} (x+2)^4-\frac {43}{6} (x+2)^3-\frac {7}{2} (x+2)^2-x \]

✓ Sympy. Time used: 0.372 (sec). Leaf size: 54

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((-x - 2)*y(x) - (-x**2 + 4*x + 5)*Derivative(y(x), x) + (x**2 + 4*x + 3)*Derivative(y(x), (x, 2)),0) 
ics = {y(-2): 2, Subs(Derivative(y(x), x), x, -2): -1} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=-2,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (x + \frac {63 \left (x + 2\right )^{4}}{8} + \frac {41 \left (x + 2\right )^{3}}{6} + \frac {7 \left (x + 2\right )^{2}}{2} + 2\right ) + C_{1} \left (- \frac {7 \left (x + 2\right )^{4}}{24} - \frac {\left (x + 2\right )^{3}}{6} + 1\right ) + O\left (x^{6}\right ) \]