9.7.8 problem problem 8

Internal problem ID [1049]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Chapter 11 Power series methods. Section 11.1 Introduction and Review of power series. Page 615
Problem number : problem 8
Date solved : Tuesday, March 04, 2025 at 12:08:04 PM
CAS classification : [_separable]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 37

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

\[ y = \left (1+\frac {1}{2} x -\frac {1}{8} x^{2}+\frac {1}{16} x^{3}-\frac {5}{128} x^{4}+\frac {7}{256} x^{5}\right ) y \left (0\right )+O\left (x^{6}\right ) \]

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

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

\[ y(x)\to c_1 \left (\frac {7 x^5}{256}-\frac {5 x^4}{128}+\frac {x^3}{16}-\frac {x^2}{8}+\frac {x}{2}+1\right ) \]

✓ Sympy. Time used: 0.725 (sec). Leaf size: 42

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)
 

\[ y{\left (x \right )} = C_{1} + \frac {C_{1} x}{2} - \frac {C_{1} x^{2}}{8} + \frac {C_{1} x^{3}}{16} - \frac {5 C_{1} x^{4}}{128} + \frac {7 C_{1} x^{5}}{256} + O\left (x^{6}\right ) \]