6.19.72 problem section 9.3, problem 72

Internal problem ID [2219]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 9 Introduction to Linear Higher Order Equations. Section 9.3. Undetermined Coefficients for Higher Order Equations. Page 495
Problem number : section 9.3, problem 72
Date solved : Tuesday, September 30, 2025 at 05:25:08 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }+2 y^{\prime \prime \prime }+2 y^{\prime \prime }+2 y^{\prime }+y&={\mathrm e}^{-x} \left (20-12 x \right ) \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=3 \\ y^{\prime }\left (0\right )&=-4 \\ y^{\prime \prime }\left (0\right )&=7 \\ y^{\prime \prime \prime }\left (0\right )&=-22 \\ \end{align*}

✓ Maple. Time used: 0.058 (sec). Leaf size: 31

ode:=diff(diff(diff(diff(y(x),x),x),x),x)+2*diff(diff(diff(y(x),x),x),x)+2*diff(diff(y(x),x),x)+2*diff(y(x),x)+y(x) = exp(-x)*(20-12*x); 
ic:=[y(0) = 3, D(y)(0) = -4, (D@@2)(y)(0) = 7, (D@@3)(y)(0) = -22]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.046 (sec). Leaf size: 39

ode=D[y[x],{x,4}]+2*D[y[x],{x,3}]+2*D[y[x],{x,2}]+2*D[y[x],x]+1*y[x]==Exp[-x]*(20-12*x); 
ic={y[0]==3,Derivative[1][y][0] ==-4,Derivative[2][y][0] ==7,Derivative[3][y][0]==-22}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{-x} \left (-x^3+2 x^2-x-e^x \sin (x)+e^x \cos (x)+2\right ) \end{align*}

✓ Sympy. Time used: 0.248 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((12*x - 20)*exp(-x) + y(x) + 2*Derivative(y(x), x) + 2*Derivative(y(x), (x, 2)) + 2*Derivative(y(x), (x, 3)) + Derivative(y(x), (x, 4)),0) 
ics = {y(0): 3, Subs(Derivative(y(x), x), x, 0): -4, Subs(Derivative(y(x), (x, 2)), x, 0): 7, Subs(Derivative(y(x), (x, 3)), x, 0): -22} 
dsolve(ode,func=y(x),ics=ics)
 

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