12.19.45 problem section 9.3, problem 45

Internal problem ID [2192]
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 45
Date solved : Monday, January 27, 2025 at 05:43:18 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }+4 y^{\prime \prime \prime }+8 y^{\prime \prime }+8 y^{\prime }+4 y&=-2 \,{\mathrm e}^{x} \left (\cos \left (x \right )-\sin \left (x \right )\right ) \end{align*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 36

dsolve(1*diff(y(x),x$4)+4*diff(y(x),x$3)+8*diff(y(x),x$2)+8*diff(y(x),x)+4*y(x)=-2*exp(x)*(cos(1*x)-sin(1*x)),y(x), singsol=all)
 

\[ y = \left (\left (x c_3 +c_1 \right ) \cos \left (x \right )+\sin \left (x \right ) \left (c_4 x +c_2 \right )\right ) {\mathrm e}^{-x}-\frac {\left (\sin \left (x \right )+\cos \left (x \right )\right ) {\mathrm e}^{x}}{16} \]

✓ Solution by Mathematica

Time used: 0.008 (sec). Leaf size: 54

DSolve[1*D[y[x],{x,4}]+4*D[y[x],{x,3}]+8*D[y[x],{x,2}]+8*D[y[x],x]+4*y[x]==-2*Exp[x]*(Cos[1*x]-Sin[1*x]),y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{16} e^{-x} \left (\left (-e^{2 x}+16 (c_4 x+c_3)\right ) \cos (x)-\left (e^{2 x}-16 (c_2 x+c_1)\right ) \sin (x)\right ) \]