81.5.4 problem 4

Internal problem ID [18596]
Book : A short course on differential equations. By Donald Francis Campbell. Maxmillan company. London. 1907
Section : Chapter IV. Ordinary linear differential equations with constant coefficients. Exercises at page 58
Problem number : 4
Date solved : Monday, March 31, 2025 at 05:46:07 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+3 y^{\prime }-y&={\mathrm e}^{x} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 31

ode:=diff(diff(y(x),x),x)+3*diff(y(x),x)-y(x) = exp(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{\frac {\left (-3+\sqrt {13}\right ) x}{2}} c_2 +{\mathrm e}^{-\frac {\left (3+\sqrt {13}\right ) x}{2}} c_1 +\frac {{\mathrm e}^{x}}{3} \]

✓ Mathematica. Time used: 0.124 (sec). Leaf size: 55

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

\[ y(x)\to \frac {1}{3} e^{-\frac {1}{2} \left (3+\sqrt {13}\right ) x} \left (e^{\frac {1}{2} \left (5+\sqrt {13}\right ) x}+3 c_2 e^{\sqrt {13} x}+3 c_1\right ) \]

✓ Sympy. Time used: 0.205 (sec). Leaf size: 34

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

\[ y{\left (x \right )} = C_{1} e^{\frac {x \left (-3 + \sqrt {13}\right )}{2}} + C_{2} e^{- \frac {x \left (3 + \sqrt {13}\right )}{2}} + \frac {e^{x}}{3} \]