45.4.5 problem 13

Internal problem ID [7287]
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. Chapter 6 review exercises. page 253
Problem number : 13
Date solved : Wednesday, March 05, 2025 at 04:22:41 AM
CAS classification : [_Laguerre]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 44

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

\[ y = c_{1} x^{3} \left (1+\frac {1}{4} x +\frac {1}{20} x^{2}+\frac {1}{120} x^{3}+\frac {1}{840} x^{4}+\frac {1}{6720} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (12+12 x +6 x^{2}+2 x^{3}+\frac {1}{2} x^{4}+\frac {1}{10} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.032 (sec). Leaf size: 66

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

\[ y(x)\to c_1 \left (\frac {x^4}{24}+\frac {x^3}{6}+\frac {x^2}{2}+x+1\right )+c_2 \left (\frac {x^7}{840}+\frac {x^6}{120}+\frac {x^5}{20}+\frac {x^4}{4}+x^3\right ) \]

✓ Sympy. Time used: 0.927 (sec). Leaf size: 31

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

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{2}}{2} + x + 1\right ) + C_{1} x^{3} \left (\frac {x^{2}}{20} + \frac {x}{4} + 1\right ) + O\left (x^{6}\right ) \]