52.2.28 problem 28

Internal problem ID [8279]
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 : 28
Date solved : Sunday, March 30, 2025 at 12:50:45 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+\frac {3 y^{\prime }}{x}-2 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.024 (sec). Leaf size: 52

Order:=8; 
ode:=diff(diff(y(x),x),x)+3/x*diff(y(x),x)-2*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \left (1+\frac {1}{4} x^{2}+\frac {1}{48} x^{4}+\frac {1}{1152} x^{6}+\operatorname {O}\left (x^{8}\right )\right ) x^{2}+c_2 \left (\ln \left (x \right ) \left (\left (-2\right ) x^{2}-\frac {1}{2} x^{4}-\frac {1}{24} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+\left (-2+\frac {3}{8} x^{4}+\frac {7}{144} x^{6}+\operatorname {O}\left (x^{8}\right )\right )\right )}{x^{2}} \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 74

ode=D[y[x],{x,2}]+3/x*D[y[x],x]-2*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_2 \left (\frac {x^6}{1152}+\frac {x^4}{48}+\frac {x^2}{4}+1\right )+c_1 \left (\frac {1}{48} \left (x^4+12 x^2+48\right ) \log (x)-\frac {5 x^6+45 x^4+72 x^2-144}{144 x^2}\right ) \]

✓ Sympy. Time used: 0.844 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-2*y(x) + Derivative(y(x), (x, 2)) + 3*Derivative(y(x), x)/x,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{1} \left (\frac {x^{6}}{1152} + \frac {x^{4}}{48} + \frac {x^{2}}{4} + 1\right ) + O\left (x^{8}\right ) \]