75.16.30 problem 503

Internal problem ID [17003]
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 : 503
Date solved : Tuesday, January 28, 2025 at 09:45:24 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }+4 y^{\prime \prime }+4 y&=\cos \left (x \right ) \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 31

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

\[ y = \left (c_{3} x +c_{1} \right ) \cos \left (\sqrt {2}\, x \right )+\left (c_4 x +c_{2} \right ) \sin \left (\sqrt {2}\, x \right )+\cos \left (x \right ) \]

✓ Solution by Mathematica

Time used: 0.257 (sec). Leaf size: 222

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

\[ y(x)\to x \cos \left (\sqrt {2} x\right ) \int _1^x-\frac {1}{4} \cos (K[2]) \cos \left (\sqrt {2} K[2]\right )dK[2]+\cos \left (\sqrt {2} x\right ) \int _1^x-\frac {1}{8} \cos (K[1]) \left (\sqrt {2} \sin \left (\sqrt {2} K[1]\right )-2 \cos \left (\sqrt {2} K[1]\right ) K[1]\right )dK[1]+\sin \left (\sqrt {2} x\right ) \int _1^x\frac {1}{8} \cos (K[3]) \left (\sqrt {2} \cos \left (\sqrt {2} K[3]\right )+2 K[3] \sin \left (\sqrt {2} K[3]\right )\right )dK[3]+x \sin \left (\sqrt {2} x\right ) \int _1^x-\frac {1}{4} \cos (K[4]) \sin \left (\sqrt {2} K[4]\right )dK[4]+c_1 \cos \left (\sqrt {2} x\right )+c_2 x \cos \left (\sqrt {2} x\right )+c_3 \sin \left (\sqrt {2} x\right )+c_4 x \sin \left (\sqrt {2} x\right ) \]