76.18.10 problem 21

Internal problem ID [17721]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 5. The Laplace transform. Section 5.2 (Properties of the Laplace transform). Problems at page 309
Problem number : 21
Date solved : Tuesday, January 28, 2025 at 11:01:49 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }-6 y&=t \,{\mathrm e}^{-t} \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=0\\ y^{\prime \prime }\left (0\right )&=0\\ y^{\prime \prime \prime }\left (0\right )&=9 \end{align*}

✓ Solution by Maple

Time used: 13.591 (sec). Leaf size: 84

dsolve([diff(y(t),t$4)-6*y(t)=t*exp(-t),y(0) = 0, D(y)(0) = 0, (D@@2)(y)(0) = 0, (D@@3)(y)(0) = 9],y(t), singsol=all)
 

\[ y = \frac {\left (-12-7 \sqrt {6}\right ) \cosh \left (6^{{1}/{4}} t \right )}{150}+\frac {\left (9 \,6^{{3}/{4}}+244 \,6^{{1}/{4}}\right ) \sinh \left (6^{{1}/{4}} t \right )}{300}+\frac {\cos \left (6^{{1}/{4}} t \right ) \left (-12+7 \sqrt {6}\right )}{150}+\frac {\left (9 \,6^{{3}/{4}}-244 \,6^{{1}/{4}}\right ) \sin \left (6^{{1}/{4}} t \right )}{300}+\frac {\left (-5 t +4\right ) {\mathrm e}^{-t}}{25} \]

✓ Solution by Mathematica

Time used: 0.012 (sec). Leaf size: 190

DSolve[{D[y[t],{t,4}]-6*y[t]==t*Exp[-t],{y[0]==0,Derivative[1][y][0] == 0,Derivative[2][y][0] == 0,Derivative[3][y][0] == 9}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {1}{600} \left (-120 e^{-t} t+96 e^{-t}-9\ 6^{3/4} e^{-\sqrt [4]{6} t}-14 \sqrt {6} e^{-\sqrt [4]{6} t}-244 \sqrt [4]{6} e^{-\sqrt [4]{6} t}-24 e^{-\sqrt [4]{6} t}+9\ 6^{3/4} e^{\sqrt [4]{6} t}-14 \sqrt {6} e^{\sqrt [4]{6} t}+244 \sqrt [4]{6} e^{\sqrt [4]{6} t}-24 e^{\sqrt [4]{6} t}+2 \sqrt [4]{6} \left (9 \sqrt {6}-244\right ) \sin \left (\sqrt [4]{6} t\right )+4 \left (7 \sqrt {6}-12\right ) \cos \left (\sqrt [4]{6} t\right )\right ) \]