83.12.7 problem 7

Internal problem ID [22008]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Chapter VII. Operational method. Ex. XV at page 121
Problem number : 7
Date solved : Thursday, October 02, 2025 at 08:21:37 PM
CAS classification : [[_3rd_order, _missing_y]]

\begin{align*} y^{\prime \prime \prime }+2 y^{\prime \prime }+y^{\prime }&={\mathrm e}^{-x} \sin \left (x \right ) \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 27

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

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

✓ Mathematica. Time used: 0.068 (sec). Leaf size: 33

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

\begin{align*} y(x)&\to \frac {1}{2} e^{-x} (\sin (x)+\cos (x)-2 (c_2 x+c_1+c_2))+c_3 \end{align*}

✓ Sympy. Time used: 0.144 (sec). Leaf size: 22

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

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