Internal problem ID [15335]
Book: A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV,
G.I. MARKARENKO. MIR, MOSCOW. 1983
Section: Chapter 2 (Higher order ODE’s). Section 15.3 Nonhomogeneous linear equations with
constant coefficients. Superposition principle. Exercises page 137
Problem number: 566.
ODE order: 4.
ODE degree: 1.
CAS Maple gives this as type [[_high_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime \prime \prime }+2 y^{\prime \prime \prime }+2 y^{\prime \prime }+2 y^{\prime }+y=x \,{\mathrm e}^{x}+\frac {\cos \left (x \right )}{2}} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 43
dsolve(diff(y(x),x$4)+2*diff(y(x),x$3)+2*diff(y(x),x$2)+2*diff(y(x),x)+y(x)=x*exp(x)+1/2*cos(x),y(x), singsol=all)
\[ y \left (x \right ) = \left (c_{4} x +c_{3} \right ) {\mathrm e}^{-x}+\frac {\left (-x +8 c_{1} +1\right ) \cos \left (x \right )}{8}+\frac {\left (x -2\right ) {\mathrm e}^{x}}{8}+\frac {\sin \left (x \right ) \left (4 c_{2} +1\right )}{4} \]
✓ Solution by Mathematica
Time used: 0.234 (sec). Leaf size: 52
DSolve[y''''[x]+2*y'''[x]+2*y''[x]+2*y'[x]+y[x]==x*Exp[x]+1/2*Cos[x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{16} \left (2 e^x (x-2)+16 e^{-x} (c_4 x+c_3)-2 (x-1-8 c_1) \cos (x)+(3+16 c_2) \sin (x)\right ) \]