75.16.12 problem 485

Internal problem ID [16985]
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 : 485
Date solved : Tuesday, January 28, 2025 at 09:44:30 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: 0.004 (sec). Leaf size: 33

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)
 

\[ y = \frac {\left (16 c_{2} {\mathrm e}^{-2 x}+{\mathrm e}^{2 x}\right ) \sin \left (2 x \right )}{16}+{\mathrm e}^{-2 x} \cos \left (2 x \right ) c_{1} \]

✓ Solution by Mathematica

Time used: 0.312 (sec). Leaf size: 85

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]
 

\[ y(x)\to \frac {1}{8} e^{-2 x} \left (8 \sin (2 x) \int _1^x\frac {1}{2} e^{4 K[1]} \cos (2 K[1]) (\cos (2 K[1])+\sin (2 K[1]))dK[1]+8 c_1 \sin (2 x)+\cos (2 x) \left (-e^{4 x} \sin ^2(2 x)+8 c_2\right )\right ) \]