75.17.29 problem 579

Internal problem ID [17078]
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 : 579
Date solved : Tuesday, January 28, 2025 at 09:50:23 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 31

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

\[ y = 1+\left (c_{1} x +c_{2} \right ) {\mathrm e}^{2 x}+x +\frac {4 \cos \left (x \right )}{25}+\frac {3 \sin \left (x \right )}{25}+\frac {\cos \left (2 x \right )}{8} \]

✓ Solution by Mathematica

Time used: 0.165 (sec). Leaf size: 77

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

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