14.20.4 problem 4

Internal problem ID [2701]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.14, The method of elimination for systems. Excercises page 258
Problem number : 4
Date solved : Monday, January 27, 2025 at 06:12:00 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=x \left (t \right )+y+{\mathrm e}^{t}\\ y^{\prime }&=x \left (t \right )-y-{\mathrm e}^{t} \end{align*}

✓ Solution by Maple

Time used: 0.050 (sec). Leaf size: 77

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

\begin{align*} x \left (t \right ) &= {\mathrm e}^{\sqrt {2}\, t} c_2 +{\mathrm e}^{-\sqrt {2}\, t} c_1 -{\mathrm e}^{t} \\ y &= \sqrt {2}\, {\mathrm e}^{\sqrt {2}\, t} c_2 -\sqrt {2}\, {\mathrm e}^{-\sqrt {2}\, t} c_1 -{\mathrm e}^{t}-{\mathrm e}^{\sqrt {2}\, t} c_2 -{\mathrm e}^{-\sqrt {2}\, t} c_1 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.383 (sec). Leaf size: 164

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

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