75.16.29 problem 502

Internal problem ID [17002]
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 : 502
Date solved : Tuesday, January 28, 2025 at 09:45:23 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }+4 y^{\prime \prime }+4 y&=\sin \left (2 x \right ) \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 35

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

\[ y = \left (c_{3} x +c_{1} \right ) \cos \left (\sqrt {2}\, x \right )+\left (c_4 x +c_{2} \right ) \sin \left (\sqrt {2}\, x \right )+\frac {\sin \left (2 x \right )}{4} \]

✓ Solution by Mathematica

Time used: 0.668 (sec). Leaf size: 272

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

\[ y(x)\to \cos \left (\sqrt {2} x\right ) \int _1^x-\frac {1}{8} \sin (2 K[1]) \left (\sqrt {2} \sin \left (\sqrt {2} K[1]\right )-2 \cos \left (\sqrt {2} K[1]\right ) K[1]\right )dK[1]+\frac {1}{32} e^{-i \left (2+\sqrt {2}\right ) x} \left (16 i \left (e^{2 i x}-e^{2 i \left (1+\sqrt {2}\right ) x}\right ) \int _1^x\frac {1}{8} \sin (2 K[2]) \left (\sqrt {2} \cos \left (\sqrt {2} K[2]\right )+2 K[2] \sin \left (\sqrt {2} K[2]\right )\right )dK[2]+4 e^{i \sqrt {2} x} x+4 e^{i \left (4+\sqrt {2}\right ) x} x+e^{2 i \left (1+\sqrt {2}\right ) x} \left (16 (c_1-i c_3)-\left (\sqrt {2}-2-16 c_2+16 i c_4\right ) x\right )+e^{2 i x} \left (\left (-\sqrt {2}+2+16 c_2+16 i c_4\right ) x+16 (c_1+i c_3)\right )\right ) \]