IB Mathematics Applications & Interpretation Question Type 2: Finding the Particular Solution to Linear Coupled Differential Equations with Some Initial Condition and Real and Distinct Eigenvalues Exercises

Solve the system of differential equations $$\begin{gathered} \frac{dx}{dt}=7x+3y, \\ \frac{dy}{dt}=2x+4y, \end{gathered}$$ subject to the initial conditions $x(0)=4$ and $y(0)=2$.

Solve the initial value problem $$\frac{dx}{dt}=2x+4y,$$ $$\frac{dy}{dt}=3x+y,$$ with initial conditions $x(0)=4$ and $y(0)=3$.

Find the particular solution of the system $$\{\begin{array}{l}\frac{dx}{dt}=5x+y\\ \frac{dy}{dt}=2x+3y\end{array}$$ with initial conditions $x(0)=4$, $y(0)=2$.

Determine $x(t)$ and $y(t)$ for $$\begin{gathered} \frac{dx}{dt}=-2x+y, \\ \frac{dy}{dt}=3x-4y, \end{gathered}$$ with $x(0)=4$, $y(0)=2$.

Find the particular solution of the system of linear differential equations: $$\{\begin{array}{l}\frac{dx}{dt}=8x+y\\ \frac{dy}{dt}=x+2y\end{array}$$ subject to the initial conditions $x(0)=4$ and $y(0)=2$.

Find the particular solution of the following system of linear differential equations: $$\frac{dx}{dt}=-x+2y$$ $$\frac{dy}{dt}=3x+4y$$ subject to the initial conditions $x(0)=4$ and $y(0)=2$.

Solve for $x(t)$ and $y(t)$ in the system $$\begin{array}{rl}\frac{dx}{dt}& =4x+2y\\ \frac{dy}{dt}& =x+3y\end{array}$$ with initial conditions $x(0)=4$ and $y(0)=2$.

Determine $x(t)$ and $y(t)$ for the following system of differential equations:

$$\{\begin{array}{l}\frac{dx}{dt}=5x+2y\\ \frac{dy}{dt}=x-y\end{array}$$

subject to the initial conditions $x(0)=4$ and $y(0)=2$.

Determine the particular solution to the system of differential equations $$\begin{array}{rl}\frac{dx}{dt}& =3x+5y\\ \frac{dy}{dt}& =x+y\end{array}$$ with initial conditions $x(0)=4$ and $y(0)=2$.

Find $x(t)$ and $y(t)$ if $$\frac{dx}{dt}=6x-y\text{and}\frac{dy}{dt}=2x+3y$$ given that $x(0)=4$ and $y(0)=2$.

Find the particular solution of the decoupled system $$\begin{gathered} \frac{dx}{dt}=x, \\ \frac{dy}{dt}=3y, \end{gathered}$$ with $x(0)=4$, $y(0)=2$.

Solve the system of differential equations $$\begin{array}{rl}\frac{dx}{dt}& =2y\\ \frac{dy}{dt}& =-x+5y\end{array}$$ with initial conditions $x(0)=4$ and $y(0)=2$.