45.3.11 problem 13

Internal problem ID [7269]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS with Modeling Applications. Dennis G. Zill. 9th edition. Brooks/Cole. CA, USA.
Section : Chapter 6. SERIES SOLUTIONS OF LINEAR EQUATIONS. Exercises. 6.3.1 page 250
Problem number : 13
Date solved : Wednesday, March 05, 2025 at 04:22:20 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} x y^{\prime \prime }+2 y^{\prime }+4 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 62

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

\[ y = \frac {c_{1} \left (1-2 x +\frac {4}{3} x^{2}-\frac {4}{9} x^{3}+\frac {4}{45} x^{4}-\frac {8}{675} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) x +c_{2} \left (\ln \left (x \right ) \left (\left (-4\right ) x +8 x^{2}-\frac {16}{3} x^{3}+\frac {16}{9} x^{4}-\frac {16}{45} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (1-12 x^{2}+\frac {112}{9} x^{3}-\frac {140}{27} x^{4}+\frac {808}{675} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x} \]

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 85

ode=x*D[y[x],{x,2}]+2*D[y[x],x]+4*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (\frac {4 x^4}{45}-\frac {4 x^3}{9}+\frac {4 x^2}{3}-2 x+1\right )+c_1 \left (\frac {4}{9} \left (4 x^3-12 x^2+18 x-9\right ) \log (x)-\frac {188 x^4-480 x^3+540 x^2-108 x-27}{27 x}\right ) \]

✓ Sympy. Time used: 0.825 (sec). Leaf size: 39

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

\[ y{\left (x \right )} = C_{1} \left (- \frac {8 x^{5}}{675} + \frac {4 x^{4}}{45} - \frac {4 x^{3}}{9} + \frac {4 x^{2}}{3} - 2 x + 1\right ) + O\left (x^{6}\right ) \]