45.1.3 problem 16 (x=0)

Internal problem ID [7203]
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 : 16 (x=0)
Date solved : Sunday, March 30, 2025 at 11:51:15 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}-2 x +10\right ) y^{\prime \prime }+x y^{\prime }-4 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 54

Order:=6; 
ode:=(x^2-2*x+10)*diff(diff(y(x),x),x)+x*diff(y(x),x)-4*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1+\frac {1}{5} x^{2}+\frac {1}{75} x^{3}+\frac {1}{750} x^{4}-\frac {13}{75000} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{20} x^{3}+\frac {1}{200} x^{4}-\frac {13}{20000} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 63

ode=(x^2-2*x+10)*D[y[x],{x,2}]+x*D[y[x],x]-4*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (-\frac {13 x^5}{20000}+\frac {x^4}{200}+\frac {x^3}{20}+x\right )+c_1 \left (-\frac {13 x^5}{75000}+\frac {x^4}{750}+\frac {x^3}{75}+\frac {x^2}{5}+1\right ) \]

✓ Sympy. Time used: 0.979 (sec). Leaf size: 39

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), x) + (x**2 - 2*x + 10)*Derivative(y(x), (x, 2)) - 4*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{4}}{750} + \frac {x^{3}}{75} + \frac {x^{2}}{5} + 1\right ) + C_{1} x \left (\frac {x^{3}}{200} + \frac {x^{2}}{20} + 1\right ) + O\left (x^{6}\right ) \]