12.13.7 problem 7

Internal problem ID [1898]
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 : 7
Date solved : Tuesday, March 04, 2025 at 01:45:51 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (4+x \right ) y^{\prime \prime }+\left (2+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 )&=2\\ y^{\prime }\left (0\right )&=5 \end{align*}

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

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

\[ y = 2+5 x -\frac {7}{4} x^{2}-\frac {3}{16} x^{3}+\frac {37}{192} x^{4}-\frac {7}{192} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

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

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

\[ y(x)\to -\frac {7 x^5}{192}+\frac {25 x^4}{192}+\frac {x^3}{16}-\frac {7 x^2}{4}+3 x+4 \]

✓ Sympy. Time used: 0.875 (sec). Leaf size: 44

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

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{4}}{192} + \frac {x^{3}}{16} - \frac {x^{2}}{4} + 1\right ) + C_{1} x \left (\frac {7 x^{3}}{192} - \frac {x^{2}}{16} - \frac {x}{4} + 1\right ) + O\left (x^{6}\right ) \]