52.1.12 problem 10

Internal problem ID [8229]
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 : 10
Date solved : Sunday, March 30, 2025 at 12:49:11 PM
CAS classification : [_Hermite]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 39

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

\[ y = \left (-x^{2}+1\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}-\frac {1}{120} x^{5}-\frac {1}{1680} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 40

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

\[ y(x)\to c_1 \left (1-x^2\right )+c_2 \left (-\frac {x^7}{1680}-\frac {x^5}{120}-\frac {x^3}{6}+x\right ) \]

✓ Sympy. Time used: 0.789 (sec). Leaf size: 27

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*Derivative(y(x), x) + 2*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)
 

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