12.19.72 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 : Monday, January 27, 2025 at 05:43:42 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*}

✓ Solution by Maple

Time used: 0.033 (sec). Leaf size: 31

dsolve([diff(y(x),x$4)+2*diff(y(x),x$3)+2*diff(y(x),x$2)+2*diff(y(x),x)+1*y(x)=exp(-x)*(20-12*x),y(0) = 3, D(y)(0) = -4, (D@@2)(y)(0) = 7, (D@@3)(y)(0) = -22],y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.074 (sec). Leaf size: 39

DSolve[{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),{y[0]==3,Derivative[1][y][0] ==-4,Derivative[2][y][0] ==7,Derivative[3][y][0]==-22}},y[x],x,IncludeSingularSolutions -> True]
 

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