14.25.7 problem 8

Internal problem ID [4012]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 11, Series Solutions to Linear Differential Equations. Exercises for 11.4. page 758
Problem number : 8
Date solved : Tuesday, September 30, 2025 at 07:00:22 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x 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: 0.024 (sec). Leaf size: 46

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

\[ y = c_1 x \left (1-\frac {1}{8} x +\operatorname {O}\left (x^{6}\right )\right )+\left (-\frac {1}{2} x +\frac {1}{16} x^{2}+\operatorname {O}\left (x^{6}\right )\right ) \ln \left (x \right ) c_2 +\left (1+\frac {1}{4} x -\frac {3}{16} x^{2}+\frac {1}{384} x^{3}+\frac {1}{18432} x^{4}+\frac {1}{737280} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \]

✓ Mathematica. Time used: 0.017 (sec). Leaf size: 52

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

\[ y(x)\to c_2 \left (x-\frac {x^2}{8}\right )+c_1 \left (\frac {x^4+48 x^3-4608 x^2+13824 x+18432}{18432}+\frac {1}{16} (x-8) x \log (x)\right ) \]

✓ Sympy. Time used: 0.225 (sec). Leaf size: 14

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

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