Internal problem ID [11348]
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: 0.062 (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],singsol=all)
\begin{align*} x \left (t \right ) &= -\frac {{\mathrm e}^{\frac {t}{3}} c_{2}}{2}-{\mathrm e}^{-\frac {t}{3}} c_{1} -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} \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.125 (sec). Leaf size: 90
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 e^{-t/3} \left (-6 e^{t/3} t-(c_1+c_2) e^{2 t/3}+2 c_1+c_2\right ) \\ y(t)\to 9 (t+1)+\frac {e^t}{2}+2 (c_1+c_2) e^{t/3}-(2 c_1+c_2) e^{-t/3} \\ \end{align*}