68.18.57 problem 63

Internal problem ID [17901]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Chapter 4 review exercises, page 219
Problem number : 63
Date solved : Thursday, October 02, 2025 at 02:29:26 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }-4 y^{\prime }+4 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 59

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

\[ y = \left (1-2 x^{2}-\frac {8}{3} x^{3}-2 x^{4}-\frac {16}{15} x^{5}\right ) y \left (0\right )+\left (x +2 x^{2}+2 x^{3}+\frac {4}{3} x^{4}+\frac {2}{3} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 62

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

\[ y(x)\to c_1 \left (-\frac {16 x^5}{15}-2 x^4-\frac {8 x^3}{3}-2 x^2+1\right )+c_2 \left (\frac {2 x^5}{3}+\frac {4 x^4}{3}+2 x^3+2 x^2+x\right ) \]

✓ Sympy. Time used: 0.220 (sec). Leaf size: 46

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

\[ y{\left (x \right )} = C_{2} \left (- 2 x^{4} - \frac {8 x^{3}}{3} - 2 x^{2} + 1\right ) + C_{1} x \left (\frac {4 x^{3}}{3} + 2 x^{2} + 2 x + 1\right ) + O\left (x^{6}\right ) \]