73.18.12 problem 27.1 (L)

Internal problem ID [15596]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 27. Differentiation and the Laplace transform. Additional Exercises. page 496
Problem number : 27.1 (L)
Date solved : Tuesday, January 28, 2025 at 08:02:58 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+13 y&=4 t +2 \,{\mathrm e}^{2 t} \sin \left (3 t \right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=4\\ y^{\prime }\left (0\right )&=3 \end{align*}

✓ Solution by Maple

Time used: 11.219 (sec). Leaf size: 47

dsolve([diff(y(t),t$2)+4*diff(y(t),t)+13*y(t)=4*t+2*exp(2*t)*sin(3*t),y(0) = 4, D(y)(0) = 3],y(t), singsol=all)
 

\[ y = -\frac {16}{169}+\frac {2 \cosh \left (2 t \right ) \left (346 \cos \left (3 t \right )+313 \sin \left (3 t \right )\right )}{169}+\frac {\left (-1423 \cos \left (3 t \right )-1226 \sin \left (3 t \right )\right ) \sinh \left (2 t \right )}{338}+\frac {4 t}{13} \]

✓ Solution by Mathematica

Time used: 0.879 (sec). Leaf size: 207

DSolve[{D[y[t],{t,2}]+4*D[y[t],t]+13*y[t]==4*t+2*Exp[2*t]*Sin[3*t],{y[0]==4,Derivative[1][y][0] ==3}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {1}{3} e^{-2 t} \left (-3 \cos (3 t) \int _1^0-\frac {2}{3} e^{2 K[2]} \sin (3 K[2]) \left (2 K[2]+e^{2 K[2]} \sin (3 K[2])\right )dK[2]+3 \cos (3 t) \int _1^t-\frac {2}{3} e^{2 K[2]} \sin (3 K[2]) \left (2 K[2]+e^{2 K[2]} \sin (3 K[2])\right )dK[2]-3 \sin (3 t) \int _1^0\frac {2}{3} e^{2 K[1]} \cos (3 K[1]) \left (2 K[1]+e^{2 K[1]} \sin (3 K[1])\right )dK[1]+3 \sin (3 t) \int _1^t\frac {2}{3} e^{2 K[1]} \cos (3 K[1]) \left (2 K[1]+e^{2 K[1]} \sin (3 K[1])\right )dK[1]+11 \sin (3 t)+12 \cos (3 t)\right ) \]