75.16.76 problem 549

Internal problem ID [17049]
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 : 549
Date solved : Tuesday, January 28, 2025 at 09:47:34 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 27

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

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

✓ Solution by Mathematica

Time used: 0.160 (sec). Leaf size: 65

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

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