52.1.16 problem 14

Internal problem ID [8233]
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. Section 6.2 SOLUTIONS ABOUT ORDINARY POINTS. EXERCISES 6.2. Page 246
Problem number : 14
Date solved : Sunday, March 30, 2025 at 12:49:16 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 43

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

\[ y = \left (1+\frac {1}{4} x^{2}-\frac {1}{24} x^{3}+\frac {1}{480} x^{5}-\frac {1}{1440} x^{6}+\frac {1}{6720} x^{7}\right ) y \left (0\right )+x y^{\prime }\left (0\right )+O\left (x^{8}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 91

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

\[ y(x)\to c_2 \left (\frac {29 x^7}{20160}-\frac {7 x^6}{1440}+\frac {x^5}{240}+\frac {x^4}{24}-\frac {x^3}{6}+x\right )+c_1 \left (\frac {x^7}{8064}+\frac {x^6}{576}-\frac {x^5}{96}+\frac {x^4}{48}+\frac {x^3}{24}-\frac {x^2}{4}+1\right ) \]

✓ Sympy. Time used: 0.773 (sec). Leaf size: 32

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

\[ y{\left (x \right )} = C_{2} \left (- \frac {x^{6}}{1440} + \frac {x^{5}}{480} - \frac {x^{3}}{24} + \frac {x^{2}}{4} + 1\right ) + C_{1} x + O\left (x^{8}\right ) \]