76.15.19 problem 20

Internal problem ID [17660]
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 : 20
Date solved : Tuesday, January 28, 2025 at 10:48:22 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }-3 y&=3 t \,{\mathrm e}^{2 t} \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.024 (sec). Leaf size: 28

dsolve([diff(y(t),t$2)-2*diff(y(t),t)-3*y(t)=3*t*exp(2*t),y(0) = 1, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = {\mathrm e}^{3 t}+\frac {2 \,{\mathrm e}^{-t}}{3}-t \,{\mathrm e}^{2 t}-\frac {2 \,{\mathrm e}^{2 t}}{3} \]

✓ Solution by Mathematica

Time used: 0.022 (sec). Leaf size: 34

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

\[ y(t)\to -\frac {1}{3} e^{2 t} (3 t+2)+\frac {2 e^{-t}}{3}+e^{3 t} \]