88.19.6 problem 6

Internal problem ID [24134]
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 139
Problem number : 6
Date solved : Thursday, October 02, 2025 at 10:00:06 PM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime }+3 k y^{\prime \prime }+3 k^{2} y^{\prime }+k^{3} y&={\mathrm e}^{-k x} f^{\prime \prime \prime }\left (x \right ) \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 22

ode:=diff(diff(diff(y(x),x),x),x)+3*k*diff(diff(y(x),x),x)+3*k^2*diff(y(x),x)+k^3*y(x) = exp(-k*x)*diff(diff(diff(f(x),x),x),x); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 26

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

\begin{align*} y(x)&\to e^{-k x} (f(x)+x (c_3 x+c_2)+c_1) \end{align*}

✓ Sympy. Time used: 1.997 (sec). Leaf size: 70

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

\[ y{\left (x \right )} = \left (C_{1} + x \left (C_{2} + x \left (C_{3} + \frac {\int e^{x} e^{k x} \frac {d^{3}}{d x^{3}} f{\left (x \right )}\, dx}{2}\right ) - \int x e^{x} e^{k x} \frac {d^{3}}{d x^{3}} f{\left (x \right )}\, dx\right ) + \frac {\int x^{2} e^{x} e^{k x} \frac {d^{3}}{d x^{3}} f{\left (x \right )}\, dx}{2}\right ) e^{- k x} \]