Internal problem ID [1537]
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.
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 }+6 y^{\prime \prime \prime }+13 y^{\prime \prime }+12 y^{\prime }+4 y-{\mathrm e}^{-x} \left (\left (-x +4\right ) \cos \relax (x )-\left (x +5\right ) \sin \relax (x )\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 65
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)
\[ y \relax (x ) = -\frac {{\mathrm e}^{-x} \cos \relax (x )}{2}+\frac {\sin \relax (x ) {\mathrm e}^{-x} x}{2}-{\mathrm e}^{-x} \sin \relax (x )-\frac {\cos \relax (x ) {\mathrm e}^{-x} x}{2}+{\mathrm e}^{-2 x} c_{1}+{\mathrm e}^{-x} c_{2}+c_{3} x \,{\mathrm e}^{-x}+c_{4} {\mathrm e}^{-2 x} x \]
✓ Solution by Mathematica
Time used: 0.021 (sec). Leaf size: 51
DSolve[1*y''''[x]+6*y'''[x]+13*y''[x]+12*y'[x]+4*y[x]==Exp[-1*x]*((4-x)*Cos[x]-(5+x)*Sin[x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} e^{-2 x} \left (2 (c_2 x+c_1)+e^x ((x-2) \sin (x)-(x+1) \cos (x)+2 (c_4 x+c_3))\right ) \\ \end{align*}