75.17.16 problem 566

Internal problem ID [17065]
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 ODEs). Section 15.3 Nonhomogeneous linear equations with constant coefficients. Superposition principle. Exercises page 137
Problem number : 566
Date solved : Tuesday, January 28, 2025 at 09:48:50 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} 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} \end{align*}

✓ Solution by Maple

Time used: 0.253 (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)
 

\[ y = \left (c_4 x +c_{3} \right ) {\mathrm e}^{-x}+\frac {\left (8 c_{1} -x +1\right ) \cos \left (x \right )}{8}+\frac {\left (x -2\right ) {\mathrm e}^{x}}{8}+\frac {\sin \left (x \right ) \left (4 c_{2} +1\right )}{4} \]

✓ Solution by Mathematica

Time used: 0.155 (sec). Leaf size: 158

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

\[ y(x)\to e^{-x} \left (e^x \cos (x) \int _1^x-\frac {1}{4} \cos (K[1]) \left (\cos (K[1])+2 e^{K[1]} K[1]\right )dK[1]+\int _1^x-\frac {1}{4} e^{K[3]} (K[3]-1) \left (\cos (K[3])+2 e^{K[3]} K[3]\right )dK[3]+x \int _1^x\frac {1}{4} e^{K[4]} \left (\cos (K[4])+2 e^{K[4]} K[4]\right )dK[4]+e^x \sin (x) \int _1^x-\frac {1}{4} \left (\cos (K[2])+2 e^{K[2]} K[2]\right ) \sin (K[2])dK[2]+c_4 x+c_1 e^x \cos (x)+c_2 e^x \sin (x)+c_3\right ) \]