75.17.30 problem 580

Internal problem ID [17079]
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. Superposition principle. Exercises page 137
Problem number : 580
Date solved : Tuesday, January 28, 2025 at 09:50:26 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+y&=1+2 \cos \left (x \right )+\cos \left (2 x \right )-\sin \left (2 x \right ) \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 30

dsolve(diff(y(x),x$2)+2*diff(y(x),x)+y(x)=1+2*cos(x)+cos(2*x)-sin(2*x),y(x), singsol=all)
 

\[ y = 1+\left (c_{1} x +c_{2} \right ) {\mathrm e}^{-x}+\sin \left (x \right )+\frac {\cos \left (2 x \right )}{25}+\frac {7 \sin \left (2 x \right )}{25} \]

✓ Solution by Mathematica

Time used: 0.904 (sec). Leaf size: 85

DSolve[D[y[x],{x,2}]+2*D[y[x],x]+y[x]==1+2*Cos[x]+Cos[2*x]-Sin[2*x],y[x],x,IncludeSingularSolutions -> True]
 

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