45.3.13 problem 15

Internal problem ID [7271]
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 : 15
Date solved : Wednesday, March 05, 2025 at 04:22:23 AM
CAS classification : [_Lienard]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 42

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

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

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 59

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

\[ y(x)\to c_1 \left (\frac {1}{16} \left (x^2-8\right ) x^2 \log (x)+\frac {1}{64} \left (-5 x^4+16 x^2+64\right )\right )+c_2 \left (\frac {x^6}{192}-\frac {x^4}{8}+x^2\right ) \]

✓ Sympy. Time used: 0.836 (sec). Leaf size: 17

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

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