Internal problem ID [10330]
Book: An elementary treatise on differential equations by Abraham Cohen. DC heath publishers.
1906
Section: Chapter X, System of simulataneous equations. Article 64. Systems of linear equations with
constant coefficients. Page 150
Problem number: Ex 1.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right )&=-x \left (t \right )-\frac {2 y \left (t \right )}{3}+\frac {{\mathrm e}^{t}}{3}\\ y^{\prime }\left (t \right )&=\frac {4 x \left (t \right )}{3}+y \left (t \right )-t \end {align*}
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 47
dsolve([3*diff(x(t),t)+3*x(t)+2*y(t)=exp(t),4*x(t)-3*diff(y(t),t)+3*y(t)=3*t],[x(t), y(t)], singsol=all)
\[ x \left (t \right ) = -{\mathrm e}^{-\frac {t}{3}} c_{2} -\frac {{\mathrm e}^{\frac {t}{3}} c_{1}}{2}-6 t \] \[ y \left (t \right ) = {\mathrm e}^{-\frac {t}{3}} c_{2} +{\mathrm e}^{\frac {t}{3}} c_{1} +9 t +9+\frac {{\mathrm e}^{t}}{2} \]
✓ Solution by Mathematica
Time used: 0.981 (sec). Leaf size: 80
DSolve[{3*x'[t]+3*x[t]+2*y[t]==Exp[t],4*x[t]-3*y'[t]+3*y[t]==3*t},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to -6 t+e^{-t/3} \left (-(c_1+c_2) e^{2 t/3}+2 c_1+c_2\right ) \\ y(t)\to c_2 \cosh \left (\frac {t}{3}\right )+\frac {1}{2} \left (18 t+\sinh (t)+\cosh (t)+(8 c_1+6 c_2) \sinh \left (\frac {t}{3}\right )+18\right ) \\ \end{align*}