74.11.43 problem 55

Internal problem ID [16293]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.3, page 156
Problem number : 55
Date solved : Tuesday, January 28, 2025 at 09:00:48 AM
CAS classification : [[_2nd_order, _missing_y]]

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

With initial conditions

\begin{align*} y \left (0\right )&=-{\frac {7}{2}}\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.023 (sec). Leaf size: 33

dsolve([diff(y(t),t$2)-diff(y(t),t)=-3*t-4*t^2*exp(2*t),y(0) = -7/2, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = -\frac {3}{2}+\left (-2 t^{2}+6 t -7\right ) {\mathrm e}^{2 t}+\frac {3 t^{2}}{2}+3 t +5 \,{\mathrm e}^{t} \]

✓ Solution by Mathematica

Time used: 7.733 (sec). Leaf size: 164

DSolve[{D[y[t],{t,2}]-D[y[t],t]==-3*t-4*t^2*Exp[2*t],{y[0]==0,Derivative[1][y][0] ==8/9}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \int _1^t\frac {1}{9} e^{K[2]} \left (-9 \int _1^0\left (-4 e^{K[1]} K[1]^2-3 e^{-K[1]} K[1]\right )dK[1]+9 \int _1^{K[2]}\left (-4 e^{K[1]} K[1]^2-3 e^{-K[1]} K[1]\right )dK[1]+8\right )dK[2]-\int _1^0\frac {1}{9} e^{K[2]} \left (-9 \int _1^0\left (-4 e^{K[1]} K[1]^2-3 e^{-K[1]} K[1]\right )dK[1]+9 \int _1^{K[2]}\left (-4 e^{K[1]} K[1]^2-3 e^{-K[1]} K[1]\right )dK[1]+8\right )dK[2] \]