Internal problem ID [274]
Book: Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section: Section 7.2, Matrices and Linear systems. Page 417
Problem number: problem 5.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\relax (t )&=2 x \relax (t )+4 y \relax (t )+3 \,{\mathrm e}^{t}\\ y^{\prime }\relax (t )&=5 x \relax (t )-y \relax (t )-t^{2} \end {align*}
✓ Solution by Maple
Time used: 0.11 (sec). Leaf size: 112
dsolve([diff(x(t),t)=2*x(t)+4*y(t)+3*exp(t),diff(y(t),t)=5*x(t)-y(t)-t^2],[x(t), y(t)], singsol=all)
\[ x \relax (t ) = \frac {{\mathrm e}^{\frac {\left (1+\sqrt {89}\right ) t}{2}} c_{2} \sqrt {89}}{10}-\frac {{\mathrm e}^{-\frac {\left (-1+\sqrt {89}\right ) t}{2}} c_{1} \sqrt {89}}{10}+\frac {3 \,{\mathrm e}^{\frac {\left (1+\sqrt {89}\right ) t}{2}} c_{2}}{10}+\frac {3 \,{\mathrm e}^{-\frac {\left (-1+\sqrt {89}\right ) t}{2}} c_{1}}{10}+\frac {2 t^{2}}{11}-\frac {3 \,{\mathrm e}^{t}}{11}-\frac {2 t}{121}+\frac {23}{1331} \] \[ y \relax (t ) = {\mathrm e}^{\frac {\left (1+\sqrt {89}\right ) t}{2}} c_{2}+{\mathrm e}^{-\frac {\left (-1+\sqrt {89}\right ) t}{2}} c_{1}-\frac {t^{2}}{11}-\frac {15 \,{\mathrm e}^{t}}{22}+\frac {12 t}{121}-\frac {17}{1331} \]
✓ Solution by Mathematica
Time used: 0.728 (sec). Leaf size: 212
DSolve[{x'[t]==2*x[t]+4*y[t]+3*Exp[t],y'[t]==5*x[t]-y[t]-t^2},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {22 t (11 t-1)+23}{1331}-\frac {3 e^t}{11}+\frac {1}{178} \left (\left (89-3 \sqrt {89}\right ) c_1-8 \sqrt {89} c_2\right ) e^{-\frac {1}{2} \left (\sqrt {89}-1\right ) t}+\frac {1}{178} \left (\left (89+3 \sqrt {89}\right ) c_1+8 \sqrt {89} c_2\right ) e^{\frac {1}{2} \left (1+\sqrt {89}\right ) t} \\ y(t)\to \frac {11 (12-11 t) t-17}{1331}-\frac {15 e^t}{22}+\left (\frac {5 c_1}{\sqrt {89}}+\frac {1}{178} \left (89-3 \sqrt {89}\right ) c_2\right ) e^{\frac {1}{2} \left (1+\sqrt {89}\right ) t}+\left (\frac {1}{178} \left (89+3 \sqrt {89}\right ) c_2-\frac {5 c_1}{\sqrt {89}}\right ) e^{-\frac {1}{2} \left (\sqrt {89}-1\right ) t} \\ \end{align*}