Question Type 1: Finding the general solution to linear coupled differential equations with real and distinct eigenvalues Exercises

Find the particular solution to the system of differential equations $\frac{dx}{dt} = 4x + y, \quad \frac{dy}{dt} = 2x + 3y$ given that $x(0) = 0$ and $y(0) = 3$.

Find the general solution to the system $\frac{dx}{dt}=7x+3y, \quad \frac{dy}{dt}=4x+2y$

Find the general solution to the system $\frac{dx}{dt} = 6x + 2y, \quad \frac{dy}{dt} = 3x + 5y$

Find the general solution to the system of differential equations: $\frac{\text{d}x}{\text{d}t} = 4x + y$ $\frac{\text{d}y}{\text{d}t} = 2x + 3y$

Find the particular solution to the system $\begin{cases} \dfrac{dx}{dt} = 2x + y \\ \dfrac{dy}{dt} = 3y \end{cases}$ given $x(0) = 4$ and $y(0) = 2$.

Find the general solution to the system of differential equations: $\frac{dx}{dt} = 2x + y, \quad \frac{dy}{dt} = 3y$

Find the particular solution to the system $\frac{dx}{dt}=x+2y, \quad \frac{dy}{dt}=4y$ given $x(0)=3$ and $y(0)=1$.

Find the particular solution to the system $\frac{dx}{dt}=6x+2y, \quad \frac{dy}{dt}=3x+5y$ given $x(0)=-1$ and $y(0)=2$.

Find the particular solution to the system of differential equations

given that $x(0) = 1$ and $y(0) = 0$.

Find the general solution to the system $\begin{cases} \dfrac{dx}{dt} = 5x + 2y \\ \dfrac{dy}{dt} = 2x + 2y \end{cases}$

Find the particular solution to the system $\frac{dx}{dt}=2x+3y, \quad \frac{dy}{dt}=-y$ given $x(0)=2$ and $y(0)=1$.

Find the general solution to the system of differential equations $\begin{cases} \dfrac{dx}{dt} = x + 2y \\ \dfrac{dy}{dt} = 4y \end{cases}$