62.24.1 problem Ex 1

Internal problem ID [12858]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter VII, Linear differential equations with constant coefficients. Article 47. Particular integral. Page 100
Problem number : Ex 1
Date solved : Monday, March 31, 2025 at 07:22:54 AM
CAS classification : [[_3rd_order, _missing_y]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 27

ode:=diff(diff(diff(y(x),x),x),x)-diff(diff(y(x),x),x)-2*diff(y(x),x) = exp(-x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{2 x} c_2}{2}+\frac {{\mathrm e}^{-x} \left (-6 c_1 +2 x +2\right )}{6}+c_3 \]

✓ Mathematica. Time used: 5.637 (sec). Leaf size: 125

ode=D[y[x],{x,3}]-D[y[x],{x,2}]-2*D[y[x],x]==Exp[-x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \int _1^x\frac {1}{9} e^{-K[1]} \left (9 c_1+9 e^{3 K[1]} c_2-3 K[1]-1\right )dK[1]+c_3 \\ y(x)\to \frac {1}{9} e^{-x} (3 x+4-9 c_1)+\frac {-\frac {7}{9}+c_1}{e}+c_3 \\ y(x)\to \frac {1}{18} \left (e^{-x} (6 x+8)+9 c_2 e^{2 x}-\frac {14}{e}-9 e^2 c_2+18 c_3\right ) \\ \end{align*}

✓ Sympy. Time used: 0.262 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-2*Derivative(y(x), x) - Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 3)) - exp(-x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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