76.15.16 problem 17

Internal problem ID [17657]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.5 (Nonhomogeneous Equations, Method of Undetermined Coefficients). Problems at page 260
Problem number : 17
Date solved : Tuesday, January 28, 2025 at 10:48:12 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+y^{\prime }-2 y&=2 t \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=1 \end{align*}

✓ Solution by Maple

Time used: 0.020 (sec). Leaf size: 29

dsolve([diff(y(t),t$2)+diff(y(t),t)-2*y(t)=2*t,y(0) = 0, D(y)(0) = 1],y(t), singsol=all)
 

\[ y = -\frac {\left (-2 \,{\mathrm e}^{3 t}+2 t \,{\mathrm e}^{2 t}+{\mathrm e}^{2 t}+1\right ) {\mathrm e}^{-2 t}}{2} \]

✓ Solution by Mathematica

Time used: 0.013 (sec). Leaf size: 24

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

\[ y(t)\to -t-\frac {e^{-2 t}}{2}+e^t-\frac {1}{2} \]