7.13.8 problem 8

Internal problem ID [407]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.1 (Introduction). Problems at page 206
Problem number : 8
Date solved : Saturday, March 29, 2025 at 04:52:58 PM
CAS classification : [_separable]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 37

Order:=6; 
ode:=2*(1+x)*diff(y(x),x)+y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {1}{2} x +\frac {3}{8} x^{2}-\frac {5}{16} x^{3}+\frac {35}{128} x^{4}-\frac {63}{256} x^{5}\right ) y \left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 41

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

\[ y(x)\to c_1 \left (-\frac {63 x^5}{256}+\frac {35 x^4}{128}-\frac {5 x^3}{16}+\frac {3 x^2}{8}-\frac {x}{2}+1\right ) \]

✓ Sympy. Time used: 0.686 (sec). Leaf size: 46

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

\[ y{\left (x \right )} = C_{1} - \frac {C_{1} x}{2} + \frac {3 C_{1} x^{2}}{8} - \frac {5 C_{1} x^{3}}{16} + \frac {35 C_{1} x^{4}}{128} - \frac {63 C_{1} x^{5}}{256} + O\left (x^{6}\right ) \]