75.17.39 problem 589

Internal problem ID [17088]
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 : 589
Date solved : Tuesday, January 28, 2025 at 09:51:58 AM
CAS classification : [[_high_order, _missing_y]]

\begin{align*} y^{\left (5\right )}-y^{\prime \prime \prime }&=x +2 \,{\mathrm e}^{-x} \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 38

dsolve(diff(y(x),x$5)-diff(y(x),x$3)=x+2*exp(-x),y(x), singsol=all)
 

\[ y = \frac {\left (7+2 x -2 c_{1} \right ) {\mathrm e}^{-x}}{2}-\frac {x^{4}}{24}+\frac {x^{2} c_{3}}{2}+c_4 x +{\mathrm e}^{x} c_{2} +c_5 \]

✓ Solution by Mathematica

Time used: 60.139 (sec). Leaf size: 113

DSolve[D[y[x],{x,5}]-D[y[x],{x,3}]==x+2*Exp[-x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \int _1^x\int _1^{K[5]}\int _1^{K[4]}e^{-K[3]} \left (e^{2 K[3]} c_1+c_2+e^{2 K[3]} \int _1^{K[3]}\frac {1}{2} e^{-2 K[1]} \left (e^{K[1]} K[1]+2\right )dK[1]+\int _1^{K[3]}\left (-\frac {1}{2} e^{K[2]} K[2]-1\right )dK[2]\right )dK[3]dK[4]dK[5]+x (c_5 x+c_4)+c_3 \]