88.22.3 problem 7

Internal problem ID [24163]
Book : Elementary Differential Equations. By Lee Roy Wilcox and Herbert J. Curtis. 1961 first edition. International texbook company. Scranton, Penn. USA. CAT number 61-15976
Section : Chapter 5. Special Techniques for Linear Equations. Exercises at page 160 (Laplace transform)
Problem number : 7
Date solved : Thursday, October 02, 2025 at 10:00:21 PM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

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

Using Laplace method

✓ Maple. Time used: 0.097 (sec). Leaf size: 80

ode:=diff(diff(diff(y(x),x),x),x)+y(x) = exp(x)*sin(x); 
dsolve(ode,y(x),method='laplace');
 

\[ y = -\frac {{\mathrm e}^{x} \left (2 \cos \left (x \right )+\sin \left (x \right )\right )}{5}+\frac {{\mathrm e}^{-x} \left (5 y \left (0\right )-5 y^{\prime }\left (0\right )+5 y^{\prime \prime }\left (0\right )+1\right )}{15}+\frac {\left (\left (1-y^{\prime \prime }\left (0\right )+2 y \left (0\right )+y^{\prime }\left (0\right )\right ) \cos \left (\frac {\sqrt {3}\, x}{2}\right )+\sin \left (\frac {\sqrt {3}\, x}{2}\right ) \sqrt {3}\, \left (y^{\prime \prime }\left (0\right )+y^{\prime }\left (0\right )+1\right )\right ) {\mathrm e}^{\frac {x}{2}}}{3} \]

✓ Mathematica. Time used: 0.321 (sec). Leaf size: 74

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

\begin{align*} y(x)&\to -\frac {1}{5} e^x \sin (x)-\frac {2}{5} e^x \cos (x)+c_1 e^{-x}+c_3 e^{x/2} \cos \left (\frac {\sqrt {3} x}{2}\right )+c_2 e^{x/2} \sin \left (\frac {\sqrt {3} x}{2}\right ) \end{align*}

✓ Sympy. Time used: 0.124 (sec). Leaf size: 51

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

\[ y{\left (x \right )} = C_{3} e^{- x} + \left (C_{1} \sin {\left (\frac {\sqrt {3} x}{2} \right )} + C_{2} \cos {\left (\frac {\sqrt {3} x}{2} \right )}\right ) e^{\frac {x}{2}} + \frac {\left (- \sin {\left (x \right )} - 2 \cos {\left (x \right )}\right ) e^{x}}{5} \]