75.16.74 problem 547

Internal problem ID [17047]
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 : 547
Date solved : Tuesday, January 28, 2025 at 09:47:33 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 27

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

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

✓ Solution by Mathematica

Time used: 0.072 (sec). Leaf size: 134

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

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