75.16.13 problem 486

Internal problem ID [16986]
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 : 486
Date solved : Tuesday, January 28, 2025 at 09:44:41 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 32

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

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

✓ Solution by Mathematica

Time used: 0.952 (sec). Leaf size: 94

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

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