75.16.28 problem 501

Internal problem ID [17001]
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 : 501
Date solved : Tuesday, January 28, 2025 at 09:45:23 AM
CAS classification : [[_high_order, _missing_y]]

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 37

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

\[ y = \frac {\left (24+x^{3}+6 x^{2}+6 \left (3+c_{1} \right ) x +12 c_{1} +6 c_{2} \right ) {\mathrm e}^{-x}}{6}+c_{3} x +c_4 \]

✓ Solution by Mathematica

Time used: 48.818 (sec). Leaf size: 168

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

\begin{align*} y(x)\to \int _1^x\int _1^{K[2]}\frac {1}{6} e^{-K[1]} \left (K[1]^3+6 c_2 K[1]+6 c_1\right )dK[1]dK[2]+c_4 x+c_3 \\ y(x)\to e^{-x} \left (\frac {x^3}{6}+x^2+3 x+4+c_1\right )+\frac {\left (\frac {8}{3}+c_1\right ) x-\frac {65}{6}-2 c_1}{e}+c_4 x+c_3 \\ y(x)\to e^{-x} \left (\frac {x^3}{6}+x^2+(3+c_2) x+2 (2+c_2)\right )+\frac {\left (\frac {8}{3}+2 c_2\right ) x-\frac {65}{6}-5 c_2}{e}+c_4 x+c_3 \\ \end{align*}