46.2.16 problem 16

Internal problem ID [9553]
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 : 16
Date solved : Tuesday, September 30, 2025 at 06:20:38 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x y^{\prime \prime }+5 y^{\prime }+x y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.027 (sec). Leaf size: 36

Order:=8; 
ode:=2*x*diff(diff(y(x),x),x)+5*diff(y(x),x)+x*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \left (1-\frac {1}{2} x^{2}+\frac {1}{40} x^{4}-\frac {1}{2160} x^{6}+\operatorname {O}\left (x^{8}\right )\right )}{x^{{3}/{2}}}+c_2 \left (1-\frac {1}{14} x^{2}+\frac {1}{616} x^{4}-\frac {1}{55440} x^{6}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 61

ode=2*x*D[y[x],{x,2}]+5*D[y[x],x]+x*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (-\frac {x^6}{55440}+\frac {x^4}{616}-\frac {x^2}{14}+1\right )+\frac {c_2 \left (-\frac {x^6}{2160}+\frac {x^4}{40}-\frac {x^2}{2}+1\right )}{x^{3/2}} \]

✓ Sympy. Time used: 0.298 (sec). Leaf size: 53

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*y(x) + 2*x*Derivative(y(x), (x, 2)) + 5*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} \left (- \frac {x^{6}}{55440} + \frac {x^{4}}{616} - \frac {x^{2}}{14} + 1\right ) + \frac {C_{1} \left (\frac {x^{8}}{224640} - \frac {x^{6}}{2160} + \frac {x^{4}}{40} - \frac {x^{2}}{2} + 1\right )}{x^{\frac {3}{2}}} + O\left (x^{8}\right ) \]