60.9.13 problem 1868

Internal problem ID [11867]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1868
Date solved : Monday, January 27, 2025 at 11:44:01 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )+3 x \left (t \right )-y \left (t \right )&={\mathrm e}^{2 t}\\ \frac {d}{d t}y \left (t \right )+x \left (t \right )+5 y \left (t \right )&={\mathrm e}^{t} \end{align*}

✓ Solution by Maple

Time used: 0.092 (sec). Leaf size: 63

dsolve([diff(x(t),t)+3*x(t)-y(t)=exp(2*t),diff(y(t),t)+x(t)+5*y(t)=exp(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= c_{2} {\mathrm e}^{-4 t}+{\mathrm e}^{-4 t} t c_{1} +\frac {{\mathrm e}^{t}}{25}+\frac {7 \,{\mathrm e}^{2 t}}{36} \\ y \left (t \right ) &= -\frac {{\mathrm e}^{2 t}}{36}-c_{2} {\mathrm e}^{-4 t}-{\mathrm e}^{-4 t} t c_{1} +\frac {4 \,{\mathrm e}^{t}}{25}+{\mathrm e}^{-4 t} c_{1} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.154 (sec). Leaf size: 166

DSolve[{D[x[t],t]+3*x[t]-y[t]==Exp[2*t],D[y[t],t]+x[t]+5*y[t]==Exp[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to e^{-4 t} \left ((t+1) \int _1^te^{5 K[1]} \left (-e^{K[1]} (K[1]-1)-K[1]\right )dK[1]+t \int _1^te^{5 K[2]} \left (e^{K[2]} K[2]+K[2]+1\right )dK[2]+c_1 t+c_2 t+c_1\right ) \\ y(t)\to e^{-4 t} \left (-t \int _1^te^{5 K[1]} \left (-e^{K[1]} (K[1]-1)-K[1]\right )dK[1]-(t-1) \int _1^te^{5 K[2]} \left (e^{K[2]} K[2]+K[2]+1\right )dK[2]+c_1 (-t)-c_2 t+c_2\right ) \\ \end{align*}