12.19.40 problem section 9.3, problem 40

Internal problem ID [2187]
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 40
Date solved : Monday, January 27, 2025 at 05:43:13 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }+6 y^{\prime \prime \prime }+13 y^{\prime \prime }+12 y^{\prime }+4 y&={\mathrm e}^{-x} \left (\left (4-x \right ) \cos \left (x \right )-\left (5+x \right ) \sin \left (x \right )\right ) \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 43

dsolve(1*diff(y(x),x$4)+6*diff(y(x),x$3)+13*diff(y(x),x$2)+12*diff(y(x),x)+4*y(x)=exp(-1*x)*((4-x)*cos(x)-(5+x)*sin(x)),y(x), singsol=all)
 

\[ y = \frac {\left (\left (-x -1\right ) \cos \left (x \right )+\sin \left (x \right ) \left (x -2\right )+2 x c_3 +2 c_2 \right ) {\mathrm e}^{-x}}{2}+{\mathrm e}^{-2 x} \left (c_4 x +c_1 \right ) \]

✓ Solution by Mathematica

Time used: 0.016 (sec). Leaf size: 56

DSolve[1*D[y[x],{x,4}]+6*D[y[x],{x,3}]+13*D[y[x],{x,2}]+12*D[y[x],x]+4*y[x]==Exp[-1*x]*((4-x)*Cos[x]-(5+x)*Sin[x]),y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{2} e^{-2 x} \left (e^x (x-2) \sin (x)-e^x (x+1) \cos (x)+2 \left (c_2 x+c_3 e^x+c_4 e^x x+c_1\right )\right ) \]