6.13.6 problem 6

Internal problem ID [1897]
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 : 6
Date solved : Tuesday, September 30, 2025 at 05:21:01 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} \left (x^{2}+3 x +3\right ) y^{\prime \prime }+\left (6+4 x \right ) y^{\prime }+2 y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=7 \\ y^{\prime }\left (0\right )&=3 \\ \end{align*}

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

Order:=6; 
ode:=(x^2+3*x+3)*diff(diff(y(x),x),x)+(6+4*x)*diff(y(x),x)+2*y(x) = 0; 
ic:=[y(0) = 7, D(y)(0) = 3]; 
dsolve([ode,op(ic)],y(x),type='series',x=0);
 

\[ y = 7+3 x -\frac {16}{3} x^{2}+\frac {13}{3} x^{3}-\frac {23}{9} x^{4}+\frac {10}{9} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 36

ode=(3+3*x+x^2)*D[y[x],{x,2}]+(6+4*x)*D[y[x],x]+2*y[x]==0; 
ic={y[0]==7,Derivative[1][y][0] ==3}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to \frac {10 x^5}{9}-\frac {23 x^4}{9}+\frac {13 x^3}{3}-\frac {16 x^2}{3}+3 x+7 \]

✓ Sympy. Time used: 0.304 (sec). Leaf size: 44

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((4*x + 6)*Derivative(y(x), x) + (x**2 + 3*x + 3)*Derivative(y(x), (x, 2)) + 2*y(x),0) 
ics = {y(0): 7, Subs(Derivative(y(x), x), x, 0): 3} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (- \frac {2 x^{4}}{9} + \frac {x^{3}}{3} - \frac {x^{2}}{3} + 1\right ) + C_{1} x \left (- \frac {x^{3}}{3} + \frac {2 x^{2}}{3} - x + 1\right ) + O\left (x^{6}\right ) \]