75.16.14 problem 487

Internal problem ID [16987]
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 : 487
Date solved : Tuesday, January 28, 2025 at 09:44:56 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+6 y^{\prime }+13 y&={\mathrm e}^{-3 x} \cos \left (2 x \right ) \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 30

dsolve(diff(y(x),x$2)+6*diff(y(x),x)+13*y(x)=exp(-3*x)*cos(2*x),y(x), singsol=all)
 

\[ y = \frac {{\mathrm e}^{-3 x} \left (\sin \left (2 x \right ) \left (x +4 c_{2} \right )+4 \left (c_{1} +\frac {1}{8}\right ) \cos \left (2 x \right )\right )}{4} \]

✓ Solution by Mathematica

Time used: 0.061 (sec). Leaf size: 70

DSolve[D[y[x],{x,2}]+6*D[y[x],x]+13*y[x]==Exp[-3*x]*Cos[2*x],y[x],x,IncludeSingularSolutions -> True]
 

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