74.11.55 problem 68

Internal problem ID [16305]
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 : 68
Date solved : Tuesday, January 28, 2025 at 09:02:15 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y^{\prime }-2 y&=f \left (t \right ) \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.322 (sec). Leaf size: 52

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

\[ y = \frac {\left (-\int _{0}^{t}f \left (\textit {\_z1} \right ) {\mathrm e}^{2 \textit {\_z1}}d \textit {\_z1} +\left (\int _{0}^{t}f \left (\textit {\_z1} \right ) {\mathrm e}^{-\textit {\_z1}}d \textit {\_z1} \right ) {\mathrm e}^{3 t}+a \left ({\mathrm e}^{3 t}-1\right )\right ) {\mathrm e}^{-2 t}}{3} \]

✓ Solution by Mathematica

Time used: 0.050 (sec). Leaf size: 123

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

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