75.17.25 problem 575

Internal problem ID [17074]
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 : 575
Date solved : Tuesday, January 28, 2025 at 09:49:29 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-4 y^{\prime }+5 y&=1+8 \cos \left (x \right )+{\mathrm e}^{2 x} \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 32

dsolve(diff(y(x),x$2)-4*diff(y(x),x)+5*y(x)=1+8*cos(x)+exp(2*x),y(x), singsol=all)
 

\[ y = {\mathrm e}^{2 x} \sin \left (x \right ) c_{2} +{\mathrm e}^{2 x} \cos \left (x \right ) c_{1} -\sin \left (x \right )+\cos \left (x \right )+\frac {1}{5}+{\mathrm e}^{2 x} \]

✓ Solution by Mathematica

Time used: 0.218 (sec). Leaf size: 89

DSolve[D[y[x],{x,2}]-4*D[y[x],x]+5*y[x]==1+8*Cos[x]+Exp[2*x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{2 x} \left (\cos (x) \int _1^x-e^{-2 K[2]} \left (8 \cos (K[2])+e^{2 K[2]}+1\right ) \sin (K[2])dK[2]+\sin (x) \int _1^xe^{-2 K[1]} \cos (K[1]) \left (8 \cos (K[1])+e^{2 K[1]}+1\right )dK[1]+c_2 \cos (x)+c_1 \sin (x)\right ) \]