56.4.70 problem 67

Internal problem ID [8959]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 67
Date solved : Monday, January 27, 2025 at 05:20:54 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right )+2 y \left (t \right )+2 t +1\\ \frac {d}{d t}y \left (t \right )&=5 x \left (t \right )+y \left (t \right )+3 t -1 \end{align*}

✓ Solution by Maple

Time used: 0.031 (sec). Leaf size: 67

dsolve([diff(x(t),t)=x(t)+2*y(t)+2*t+1,diff(y(t),t)=5*x(t)+y(t)+3*t-1],singsol=all)
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{\left (1+\sqrt {10}\right ) t} c_{2} +{\mathrm e}^{-\left (-1+\sqrt {10}\right ) t} c_{1} -\frac {4 t}{9}+\frac {17}{81} \\ y \left (t \right ) &= \frac {{\mathrm e}^{\left (1+\sqrt {10}\right ) t} c_{2} \sqrt {10}}{2}-\frac {{\mathrm e}^{-\left (-1+\sqrt {10}\right ) t} c_{1} \sqrt {10}}{2}-\frac {7 t}{9}-\frac {67}{81} \\ \end{align*}

✓ Solution by Mathematica

Time used: 11.167 (sec). Leaf size: 158

DSolve[{D[x[t],t]==x[t]+2*y[t]+2*t+1,D[y[t],t]==5*x[t]+y[t]+3*t-1},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{810} e^{t-\sqrt {10} t} \left (e^{\left (\sqrt {10}-1\right ) t} (170-360 t)+81 \left (5 c_1+\sqrt {10} c_2\right ) e^{2 \sqrt {10} t}+81 \left (5 c_1-\sqrt {10} c_2\right )\right ) \\ y(t)\to \frac {1}{324} e^{t-\sqrt {10} t} \left (-4 e^{\left (\sqrt {10}-1\right ) t} (63 t+67)+81 \left (\sqrt {10} c_1+2 c_2\right ) e^{2 \sqrt {10} t}-81 \left (\sqrt {10} c_1-2 c_2\right )\right ) \\ \end{align*}