12.20.16 problem section 9.4, problem 39

Internal problem ID [2237]
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 39
Date solved : Tuesday, March 04, 2025 at 01:52:13 PM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }-5 y^{\prime \prime }+4 y&=F \left (x \right ) \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 84

ode:=diff(diff(diff(diff(y(x),x),x),x),x)-5*diff(diff(y(x),x),x)+4*y(x) = F(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {\left (\left (\int {\mathrm e}^{-x} F \left (x \right )d x \right ) {\mathrm e}^{3 x}-6 c_1 \,{\mathrm e}^{3 x}-\frac {\left (\int F \left (x \right ) {\mathrm e}^{-2 x}d x \right ) {\mathrm e}^{4 x}}{2}-6 c_4 \,{\mathrm e}^{4 x}-\left (\int {\mathrm e}^{x} F \left (x \right )d x \right ) {\mathrm e}^{x}-6 c_3 \,{\mathrm e}^{x}+\frac {\left (\int F \left (x \right ) {\mathrm e}^{2 x}d x \right )}{2}-6 c_2 \right ) {\mathrm e}^{-2 x}}{6} \]

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 130

ode=D[y[x],{x,4}]-5*D[y[x],{x,2}]+4*y[x]==f[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to e^{-2 x} \left (\int _1^x-\frac {1}{12} e^{2 K[1]} f(K[1])dK[1]+e^x \int _1^x\frac {1}{6} e^{K[2]} f(K[2])dK[2]+e^{3 x} \int _1^x-\frac {1}{6} e^{-K[3]} f(K[3])dK[3]+e^{4 x} \int _1^x\frac {1}{12} e^{-2 K[4]} f(K[4])dK[4]+c_2 e^x+c_3 e^{3 x}+c_4 e^{4 x}+c_1\right ) \]

✓ Sympy. Time used: 1.262 (sec). Leaf size: 70

from sympy import * 
x = symbols("x") 
y = Function("y") 
F = Function("F") 
ode = Eq(-F(x) + 4*y(x) - 5*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 4)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (C_{1} - \frac {\int F{\left (x \right )} e^{- x}\, dx}{6}\right ) e^{x} + \left (C_{2} - \frac {\int F{\left (x \right )} e^{2 x}\, dx}{12}\right ) e^{- 2 x} + \left (C_{3} + \frac {\int F{\left (x \right )} e^{- 2 x}\, dx}{12}\right ) e^{2 x} + \left (C_{4} + \frac {\int F{\left (x \right )} e^{x}\, dx}{6}\right ) e^{- x} \]