52.4.4 problem 12

Internal problem ID [8314]
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. CHAPTER 6 IN REVIEW. Page 271
Problem number : 12
Date solved : Wednesday, March 05, 2025 at 05:34:52 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 28

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

\[ y = \left (1-\frac {1}{6} x^{3}-\frac {1}{90} x^{6}\right ) y \left (0\right )+y^{\prime }\left (0\right ) x +O\left (x^{8}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 27

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

\[ y(x)\to c_1 \left (-\frac {x^6}{90}-\frac {x^3}{6}+1\right )+c_2 x \]

✓ Sympy. Time used: 0.758 (sec). Leaf size: 22

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

\[ y{\left (x \right )} = C_{2} \left (- \frac {x^{6}}{90} - \frac {x^{3}}{6} + 1\right ) + C_{1} x + O\left (x^{8}\right ) \]