12.13.42 problem 41

Internal problem ID [1933]
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 : 41
Date solved : Tuesday, March 04, 2025 at 01:46:26 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+\left (x^{2}+2 x +1\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 )&=2\\ y^{\prime }\left (0\right )&=3 \end{align*}

✓ Maple. Time used: 0.000 (sec). Leaf size: 20

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

\[ y = 2+3 x -\frac {7}{2} x^{2}-\frac {5}{6} x^{3}+\frac {41}{24} x^{4}+\frac {41}{120} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

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

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

\[ y(x)\to \frac {41 x^5}{120}+\frac {41 x^4}{24}-\frac {5 x^3}{6}-\frac {7 x^2}{2}+3 x+2 \]

✓ Sympy. Time used: 0.876 (sec). Leaf size: 44

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

\[ y{\left (x \right )} = C_{2} \left (\frac {5 x^{4}}{12} + \frac {x^{3}}{3} - x^{2} + 1\right ) + C_{1} x \left (\frac {7 x^{3}}{24} - \frac {x^{2}}{2} - \frac {x}{2} + 1\right ) + O\left (x^{6}\right ) \]