12.20.14 problem section 9.4, problem 35

Internal problem ID [2235]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 9 Introduction to Linear Higher Order Equations. Section 9.4. Variation of Parameters for Higher Order Equations. Page 503
Problem number : section 9.4, problem 35
Date solved : Tuesday, March 04, 2025 at 01:52:11 PM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 62

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

\[ y = \frac {\left (\left (\int {\mathrm e}^{-x} F \left (x \right )d x \right ) {\mathrm e}^{3 x}+6 c_1 \,{\mathrm e}^{3 x}-3 \left (\int {\mathrm e}^{x} F \left (x \right )d x \right ) {\mathrm e}^{x}+6 c_3 \,{\mathrm e}^{x}+2 \left (\int F \left (x \right ) {\mathrm e}^{2 x}d x \right )+6 c_2 \right ) {\mathrm e}^{-2 x}}{6} \]

✓ Mathematica. Time used: 0.041 (sec). Leaf size: 96

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

\[ y(x)\to e^{-2 x} \left (\int _1^x\frac {1}{3} e^{2 K[1]} f(K[1])dK[1]+e^x \int _1^x-\frac {1}{2} e^{K[2]} f(K[2])dK[2]+e^{3 x} \int _1^x\frac {1}{6} e^{-K[3]} f(K[3])dK[3]+c_2 e^x+c_3 e^{3 x}+c_1\right ) \]

✓ Sympy. Time used: 1.285 (sec). Leaf size: 51

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

\[ y{\left (x \right )} = \left (C_{1} - \frac {\int F{\left (x \right )} e^{x}\, dx}{2}\right ) e^{- x} + \left (C_{2} + \frac {\int F{\left (x \right )} e^{- x}\, dx}{6}\right ) e^{x} + \left (C_{3} + \frac {\int F{\left (x \right )} e^{2 x}\, dx}{3}\right ) e^{- 2 x} \]