75.16.58 problem 531

Internal problem ID [17031]
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 : 531
Date solved : Tuesday, January 28, 2025 at 09:46:46 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 28

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

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

✓ Solution by Mathematica

Time used: 0.050 (sec). Leaf size: 62

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

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