76.6.13 problem 15

Internal problem ID [17468]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 3. Systems of two first order equations. Section 3.2 (Two first order linear differential equations). Problems at page 142
Problem number : 15
Date solved : Tuesday, January 28, 2025 at 10:39:09 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-x \left (t \right )+y \left (t \right )+1\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )+y \left (t \right )-3 \end{align*}

✓ Solution by Maple

Time used: 0.068 (sec). Leaf size: 69

dsolve([diff(x(t),t)=-x(t)+y(t)+1,diff(y(t),t)=x(t)+y(t)-3],singsol=all)
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{\sqrt {2}\, t} c_{2} +{\mathrm e}^{-\sqrt {2}\, t} c_{1} +2 \\ y \left (t \right ) &= \sqrt {2}\, {\mathrm e}^{\sqrt {2}\, t} c_{2} -\sqrt {2}\, {\mathrm e}^{-\sqrt {2}\, t} c_{1} +{\mathrm e}^{\sqrt {2}\, t} c_{2} +{\mathrm e}^{-\sqrt {2}\, t} c_{1} +1 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.215 (sec). Leaf size: 160

DSolve[{D[x[t],t]==-x[t]+y[t]+1,D[y[t],t]==x[t]+y[t]-3},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{4} e^{-\sqrt {2} t} \left (8 e^{\sqrt {2} t}+\left (\sqrt {2} c_2-\left (\sqrt {2}-2\right ) c_1\right ) e^{2 \sqrt {2} t}+\left (2+\sqrt {2}\right ) c_1-\sqrt {2} c_2\right ) \\ y(t)\to \frac {1}{4} e^{-\sqrt {2} t} \left (4 e^{\sqrt {2} t}+\left (\sqrt {2} c_1+\left (2+\sqrt {2}\right ) c_2\right ) e^{2 \sqrt {2} t}-\sqrt {2} c_1-\left (\sqrt {2}-2\right ) c_2\right ) \\ \end{align*}