Internal problem ID [1542]
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.
ODE order: 4.
ODE degree: 1.
CAS Maple gives this as type [[_high_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime \prime \prime }+4 y^{\prime \prime \prime }+8 y^{\prime \prime }+8 y^{\prime }+4 y+2 \left (\cos \relax (x )-\sin \relax (x )\right ) {\mathrm e}^{x}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.027 (sec). Leaf size: 51
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)
\[ y \relax (x ) = -\frac {\cos \relax (x ) {\mathrm e}^{x}}{16}-\frac {\sin \relax (x ) {\mathrm e}^{x}}{16}+c_{1} \cos \relax (x ) {\mathrm e}^{-x}+c_{2} \sin \relax (x ) {\mathrm e}^{-x}+c_{3} {\mathrm e}^{-x} \cos \relax (x ) x +c_{4} {\mathrm e}^{-x} \sin \relax (x ) x \]
✓ Solution by Mathematica
Time used: 0.175 (sec). Leaf size: 45
DSolve[1*y''''[x]+4*y'''[x]+8*y''[x]+8*y'[x]+4*y[x]==-2*Exp[x]*(Cos[1*x]-Sin[1*x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {1}{16} e^x (\sin (x)+\cos (x))+e^{-x} ((c_4 x+c_3) \cos (x)+(c_2 x+c_1) \sin (x)) \\ \end{align*}