The solution of the system $\frac { d x } { d t } = - 3 x + 2 y , \frac { d y } { d t } = 4 x - y$ is
A) $x = c _ { 1 } e ^ { t } + c _ { 2 } e ^ { - 5 t } , y = 2 c _ { 1 } e ^ { t } - c _ { 2 } e ^ { - 5 t }$
B) $x = c _ { 1 } e ^ { t } + c _ { 2 } e ^ { 5 t } , y = 2 c _ { 1 } e ^ { t } - c _ { 2 } e ^ { 5 t }$
C) $x = 2 c _ { 1 } e ^ { t } + c _ { 2 } e ^ { - 5 t } , y = c _ { 1 } e ^ { t } - c _ { 2 } e ^ { - 5 t }$
D) $x = 2 c _ { 1 } e ^ { t } + c _ { 2 } e ^ { 5 t } , y = c _ { 1 } e ^ { t } - c _ { 2 } e ^ { 5 t }$
E) $x = 2 c _ { 1 } e ^ { t } + c _ { 2 } e ^ { - 5 t } , y = 2 c _ { 1 } e ^ { t } - c _ { 2 } e ^ { - 5 t }$

Question

Quizplus · Accepted Answer

The given system of differential equations can be solved using the method of eigenvalues and eigenvectors. The characteristic equation of the system's matrix leads to eigenvalues $1$ and $-5$, with corresponding eigenvectors that lead to the solution form given in option A.

The Solution of the System dxdt=−3x+2y,dydt=4x−y\frac { d x } { d t } = - 3 x + 2 y , \frac { d y } { d t } = 4 x - ydtdx​=−3x+2y,dtdy​=4x−y

The Solution of the System $\frac { d x } { d t } = - 3 x + 2 y , \frac { d y } { d t } = 4 x - y$