75.16.27 problem 500

Internal problem ID [17000]
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. Trial and error method. Exercises page 132
Problem number : 500
Date solved : Tuesday, January 28, 2025 at 09:45:22 AM
CAS classification : [[_high_order, _missing_y]]

\begin{align*} y^{\prime \prime \prime \prime }+2 y^{\prime \prime \prime }+y^{\prime \prime }&={\mathrm e}^{-x} \end{align*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 33

dsolve(diff(y(x),x$4)+2*diff(y(x),x$3)+diff(y(x),x$2)=exp(-x),y(x), singsol=all)
 

\[ y = \frac {\left (x^{2}+\left (2 c_{1} +4\right ) x +4 c_{1} +2 c_{2} +6\right ) {\mathrm e}^{-x}}{2}+c_{3} x +c_4 \]

✓ Solution by Mathematica

Time used: 40.487 (sec). Leaf size: 158

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

\begin{align*} y(x)\to \int _1^x\int _1^{K[2]}\frac {1}{2} e^{-K[1]} (2 c_1+K[1] (2 c_2+K[1]))dK[1]dK[2]+c_4 x+c_3 \\ y(x)\to e^{-x} \left (\frac {x^2}{2}+2 x+3+c_1\right )+\frac {\left (\frac {5}{2}+c_1\right ) x-2 (4+c_1)}{e}+c_4 x+c_3 \\ y(x)\to \frac {1}{2} e^{-x} \left (x^2+2 (2+c_2) x+6+4 c_2\right )+\frac {\left (\frac {5}{2}+2 c_2\right ) x-8-5 c_2}{e}+c_4 x+c_3 \\ \end{align*}