74.16.7 problem 7

Internal problem ID [16442]
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 : 7
Date solved : Monday, March 31, 2025 at 02:53:28 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-y^{\prime }-2 y&={\mathrm e}^{-x} \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 72

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

\[ y = \left (1+x^{2}+\frac {1}{3} x^{3}+\frac {1}{4} x^{4}+\frac {1}{12} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{2}+\frac {1}{2} x^{3}+\frac {5}{24} x^{4}+\frac {11}{120} x^{5}\right ) y^{\prime }\left (0\right )+\frac {x^{2}}{2}+\frac {x^{4}}{8}+\frac {x^{5}}{60}+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 87

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

\[ y(x)\to \frac {x^5}{60}+\frac {x^4}{8}+\frac {x^2}{2}+c_1 \left (\frac {x^5}{12}+\frac {x^4}{4}+\frac {x^3}{3}+x^2+1\right )+c_2 \left (\frac {11 x^5}{120}+\frac {5 x^4}{24}+\frac {x^3}{2}+\frac {x^2}{2}+x\right ) \]

✗ Sympy

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

ValueError : ODE -2*y(x) - Derivative(y(x), x) + Derivative(y(x), (x, 2)) - exp(-x) does not match hint 2nd_power_series_regular