45.1.19 problem 31

Internal problem ID [7219]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS with Modeling Applications. Dennis G. Zill. 9th edition. Brooks/Cole. CA, USA.
Section : Chapter 6. SERIES SOLUTIONS OF LINEAR EQUATIONS. Exercises. 6.1.2 page 230
Problem number : 31
Date solved : Sunday, March 30, 2025 at 11:51:35 AM
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: 14

Order:=6; 
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);
 

\[ y = 3-12 x^{2}+4 x^{4}+\operatorname {O}\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 22

ode=D[y[x],{x,2}]-2*x*D[y[x],{x,2}]+8*y[x]==0; 
ic={y[0]==3,Derivative[1][y][0] ==0}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to \frac {16 x^5}{5}-8 x^3-12 x^2+3 \]

✓ Sympy. Time used: 0.818 (sec). Leaf size: 29

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=6)
 

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