75.16.25 problem 498

Internal problem ID [16998]
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 : 498
Date solved : Tuesday, January 28, 2025 at 09:45:21 AM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime \prime }+4 y^{\prime \prime \prime }+4 y^{\prime \prime }&=1 \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 31

dsolve(diff(y(x),x$4)+4*diff(y(x),x$3)+4*diff(y(x),x$2)=1,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 28.187 (sec). Leaf size: 133

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

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