45.3.12 problem 14

Internal problem ID [7270]
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 : 14
Date solved : Sunday, March 30, 2025 at 11:53:01 AM
CAS classification : [_Lienard]

\begin{align*} x y^{\prime \prime }+3 y^{\prime }+x y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.029 (sec). Leaf size: 46

Order:=6; 
ode:=x*diff(diff(y(x),x),x)+3*diff(y(x),x)+x*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \left (1-\frac {1}{8} x^{2}+\frac {1}{192} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) x^{2}+c_2 \left (\ln \left (x \right ) \left (x^{2}-\frac {1}{8} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2+\frac {3}{32} x^{4}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{2}} \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 57

ode=x*D[y[x],{x,2}]+3*D[y[x],x]+x*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (\frac {x^4}{192}-\frac {x^2}{8}+1\right )+c_1 \left (\frac {1}{16} \left (x^2-8\right ) \log (x)-\frac {5 x^4-16 x^2-64}{64 x^2}\right ) \]

✓ Sympy. Time used: 0.820 (sec). Leaf size: 19

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

\[ y{\left (x \right )} = C_{1} \left (\frac {x^{4}}{192} - \frac {x^{2}}{8} + 1\right ) + O\left (x^{6}\right ) \]