75.18.20 problem 609

Internal problem ID [17108]
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. Initial value problem. Exercises page 140
Problem number : 609
Date solved : Tuesday, January 28, 2025 at 09:52:58 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+5 y&=4 \cos \left (2 x \right )+\sin \left (2 x \right ) \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 29

dsolve(diff(y(x),x$2)+2*diff(y(x),x)+5*y(x)=4*cos(2*x)+sin(2*x),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.334 (sec). Leaf size: 83

DSolve[D[y[x],{x,2}]+2*D[y[x],x]+5*y[x]==4*Cos[2*x]+Sin[2*x],y[x],x,IncludeSingularSolutions -> True]
 

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