76.15.18 problem 19

Internal problem ID [17659]
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 : 19
Date solved : Tuesday, January 28, 2025 at 10:48:19 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }+y&=t \,{\mathrm e}^{t}+4 \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.026 (sec). Leaf size: 18

dsolve([diff(y(t),t$2)-2*diff(y(t),t)+y(t)=t*exp(t)+4,y(0) = 1, D(y)(0) = 1],y(t), singsol=all)
 

\[ y = 4+\frac {\left (t^{3}+24 t -18\right ) {\mathrm e}^{t}}{6} \]

✓ Solution by Mathematica

Time used: 0.250 (sec). Leaf size: 23

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

\[ y(t)\to e^t \left (\frac {t^3}{6}+4 t-3\right )+4 \]