68.16.4 problem 4

Internal problem ID [17797]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.8, page 203
Problem number : 4
Date solved : Thursday, October 02, 2025 at 02:28:20 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+3 y^{\prime }-18 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 59

Order:=6; 
ode:=diff(diff(y(x),x),x)+3*diff(y(x),x)-18*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1+9 x^{2}-9 x^{3}+\frac {81}{4} x^{4}-\frac {81}{4} x^{5}\right ) y \left (0\right )+\left (x -\frac {3}{2} x^{2}+\frac {9}{2} x^{3}-\frac {45}{8} x^{4}+\frac {297}{40} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 66

ode=D[y[x],{x,2}]+3*D[y[x],x]-18*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (-\frac {81 x^5}{4}+\frac {81 x^4}{4}-9 x^3+9 x^2+1\right )+c_2 \left (\frac {297 x^5}{40}-\frac {45 x^4}{8}+\frac {9 x^3}{2}-\frac {3 x^2}{2}+x\right ) \]

✓ Sympy. Time used: 0.238 (sec). Leaf size: 49

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-18*y(x) + 3*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)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {81 x^{4}}{4} - 9 x^{3} + 9 x^{2} + 1\right ) + C_{1} x \left (- \frac {45 x^{3}}{8} + \frac {9 x^{2}}{2} - \frac {3 x}{2} + 1\right ) + O\left (x^{6}\right ) \]