52.2.25 problem 25

Internal problem ID [8276]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 6 SERIES SOLUTIONS OF LINEAR EQUATIONS. 6.3 SOLUTIONS ABOUT SINGULAR POINTS. EXERCISES 6.3. Page 255
Problem number : 25
Date solved : Sunday, March 30, 2025 at 12:50:39 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }+2 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: 36

Order:=8; 
ode:=x*diff(diff(y(x),x),x)+2*diff(y(x),x)-x*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \left (1+\frac {1}{6} x^{2}+\frac {1}{120} x^{4}+\frac {1}{5040} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+\frac {c_2 \left (1+\frac {1}{2} x^{2}+\frac {1}{24} x^{4}+\frac {1}{720} x^{6}+\operatorname {O}\left (x^{8}\right )\right )}{x} \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 56

ode=x*D[y[x],{x,2}]+2*D[y[x],x]-x*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (\frac {x^5}{720}+\frac {x^3}{24}+\frac {x}{2}+\frac {1}{x}\right )+c_2 \left (\frac {x^6}{5040}+\frac {x^4}{120}+\frac {x^2}{6}+1\right ) \]

✓ Sympy. Time used: 0.911 (sec). Leaf size: 49

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*y(x) + x*Derivative(y(x), (x, 2)) + 2*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{6}}{5040} + \frac {x^{4}}{120} + \frac {x^{2}}{6} + 1\right ) + \frac {C_{1} \left (\frac {x^{8}}{40320} + \frac {x^{6}}{720} + \frac {x^{4}}{24} + \frac {x^{2}}{2} + 1\right )}{x} + O\left (x^{8}\right ) \]