76.15.17 problem 18

Internal problem ID [17658]
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 : 18
Date solved : Tuesday, January 28, 2025 at 10:48:14 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=t^{2}+3 \,{\mathrm e}^{t} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=2 \end{align*}

✓ Solution by Maple

Time used: 0.021 (sec). Leaf size: 27

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

\[ y = \frac {7 \sin \left (2 t \right )}{10}-\frac {19 \cos \left (2 t \right )}{40}+\frac {t^{2}}{4}-\frac {1}{8}+\frac {3 \,{\mathrm e}^{t}}{5} \]

✓ Solution by Mathematica

Time used: 0.195 (sec). Leaf size: 33

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

\[ y(t)\to \frac {1}{40} \left (10 t^2+24 e^t+28 \sin (2 t)-19 \cos (2 t)-5\right ) \]