52.1.23 problem 21

Internal problem ID [8240]
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 : 21
Date solved : Sunday, March 30, 2025 at 12:49:26 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-2 x y^{\prime }+8 y&=0 \end{align*}

Using series method with expansion around

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

With initial conditions

\begin{align*} y \left (0\right )&=3\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 16

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

\[ y = 4 x^{4}-12 x^{2}+3 \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 15

ode=D[y[x],{x,2}]-2*x*D[y[x],x]+8*y[x]==0; 
ic={y[0]==3,Derivative[1][y][0] ==0}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to 4 x^4-12 x^2+3 \]

✓ Sympy. Time used: 0.793 (sec). Leaf size: 34

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

\[ y{\left (x \right )} = C_{2} \left (\frac {4 x^{4}}{3} - 4 x^{2} + 1\right ) + C_{1} x \left (\frac {x^{4}}{10} - x^{2} + 1\right ) + O\left (x^{8}\right ) \]