Internal problem ID [6538]
Book: Own collection of miscellaneous problems
Section: section 4.0
Problem number: 67.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\relax (t )&=x \relax (t )+2 y \relax (t )+2 t +1\\ y^{\prime }\relax (t )&=5 x \relax (t )+y \relax (t )+3 t -1 \end {align*}
✓ Solution by Maple
Time used: 0.074 (sec). Leaf size: 68
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],[x(t), y(t)], singsol=all)
\[ x \relax (t ) = \frac {{\mathrm e}^{\left (1+\sqrt {10}\right ) t} c_{2} \sqrt {10}}{5}-\frac {{\mathrm e}^{-\left (-1+\sqrt {10}\right ) t} c_{1} \sqrt {10}}{5}-\frac {4 t}{9}+\frac {17}{81} \] \[ y \relax (t ) = {\mathrm e}^{\left (1+\sqrt {10}\right ) t} c_{2}+{\mathrm e}^{-\left (-1+\sqrt {10}\right ) t} c_{1}-\frac {7 t}{9}-\frac {67}{81} \]
✓ Solution by Mathematica
Time used: 1.492 (sec). Leaf size: 342
DSolve[{x'[t]==x[t]+2*y[t]+2*t+1,y'[t]==5*x[t]+y[t]+3*t-1},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \sqrt {\frac {2}{5}} e^t \sinh \left (\sqrt {10} t\right ) \left (\int _1^t\frac {1}{4} e^{-\left (\left (1+\sqrt {10}\right ) K[2]\right )} \left (2 \left (3+\sqrt {10}\right ) K[2]-e^{2 \sqrt {10} K[2]} \left (2 \left (-3+\sqrt {10}\right ) K[2]+\sqrt {10}+2\right )+\sqrt {10}-2\right )dK[2]+c_2\right )+e^t \cosh \left (\sqrt {10} t\right ) \left (\int _1^t\frac {1}{5} e^{-K[1]} \left (5 \cosh \left (\sqrt {10} K[1]\right ) (2 K[1]+1)+\sqrt {10} (1-3 K[1]) \sinh \left (\sqrt {10} K[1]\right )\right )dK[1]+c_1\right ) \\ y(t)\to e^t \cosh \left (\sqrt {10} t\right ) \left (\int _1^t\frac {1}{4} e^{-\left (\left (1+\sqrt {10}\right ) K[2]\right )} \left (2 \left (3+\sqrt {10}\right ) K[2]-e^{2 \sqrt {10} K[2]} \left (2 \left (-3+\sqrt {10}\right ) K[2]+\sqrt {10}+2\right )+\sqrt {10}-2\right )dK[2]+c_2\right )+\sqrt {\frac {5}{2}} e^t \sinh \left (\sqrt {10} t\right ) \left (\int _1^t\frac {1}{5} e^{-K[1]} \left (5 \cosh \left (\sqrt {10} K[1]\right ) (2 K[1]+1)+\sqrt {10} (1-3 K[1]) \sinh \left (\sqrt {10} K[1]\right )\right )dK[1]+c_1\right ) \\ \end{align*}