71.12.3 problem 3

Internal problem ID [14448]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 4. N-th Order Linear Differential Equations. Exercises 4.5, page 221
Problem number : 3
Date solved : Monday, March 31, 2025 at 12:27:25 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

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

With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=3\\ y^{\prime \prime }\left (0\right )&=-3 \end{align*}

✓ Maple. Time used: 0.042 (sec). Leaf size: 19

ode:=diff(diff(diff(y(x),x),x),x)-diff(diff(y(x),x),x)+diff(y(x),x)-y(x) = 2*exp(x); 
ic:=y(0) = 1, D(y)(0) = 3, (D@@2)(y)(0) = -3; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = \left (x -2\right ) {\mathrm e}^{x}+3 \cos \left (x \right )+4 \sin \left (x \right ) \]

✓ Mathematica. Time used: 0.017 (sec). Leaf size: 21

ode=D[y[x],{x,3}]-D[y[x],{x,2}]+D[y[x],x]-y[x]==2*Exp[x]; 
ic={y[0]==1,Derivative[1][y][0] ==3,Derivative[2][y][0] ==-3}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to e^x (x-2)+4 \sin (x)+3 \cos (x) \]

✓ Sympy. Time used: 0.189 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-y(x) - 2*exp(x) + Derivative(y(x), x) - Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 3)),0) 
ics = {y(0): 1, Subs(Derivative(y(x), x), x, 0): 3, Subs(Derivative(y(x), (x, 2)), x, 0): -3} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (x - 2\right ) e^{x} + 4 \sin {\left (x \right )} + 3 \cos {\left (x \right )} \]