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Exercises for Question Type 2: Finding the particular solution to linear coupled differential equations with some initial condition and real and distinct eigenvalues - IB | RevisionDojo

Question 1

Solve the system of differential equations

\frac{dx}{dt}=7x+3y, \\ \frac{dy}{dt}=2x+4y,

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

[7]

Question 2

Solve the initial value problem

\frac{dx}{dt} = 2x + 4y,

\frac{dy}{dt} = 3x + y,

with initial conditions $x(0) = 4$ and $y(0) = 3$ .

[7]

Question 3

Find the particular solution of the system

\begin{cases} \dfrac{dx}{dt} = 5x + y \\ \dfrac{dy}{dt} = 2x + 3y \end{cases}

with initial conditions $x(0) = 4$ , $y(0) = 2$ .

[10]

Question 4

Determine $x(t)$ and $y(t)$ for

\frac{dx}{dt}=-2x+y, \\ \frac{dy}{dt}=3x-4y,

with $x(0)=4$ , $y(0)=2$ .

[9]

Question 5

Systems of Differential Equations

Find the particular solution of the system of linear differential equations:

\begin{cases} \frac{dx}{dt} = 8x + y \\ \frac{dy}{dt} = x + 2y \end{cases}

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

[7]

Question 6

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 $y(0) = 2$ .

[8]

Question 7

Solve for $x(t)$ and $y(t)$ in the system

\begin{aligned} \frac{dx}{dt} &= 4x + 2y \\ \frac{dy}{dt} &= x + 3y \end{aligned}

with initial conditions $x(0) = 4$ and $y(0) = 2$ .

[7]

Question 8

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

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

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

[8]

Question 9

Determine the particular solution to the system of differential equations

\begin{aligned} \frac{dx}{dt} &= 3x + 5y \\ \frac{dy}{dt} &= x + y \end{aligned}

with initial conditions $x(0) = 4$ and $y(0) = 2$ .

[8]

Question 10

The problem asks for the solution of a system of first-order linear differential equations with given initial conditions.

Find $x(t)$ and $y(t)$ if

\frac{dx}{dt} = 6x - y \quad \text{and} \quad \frac{dy}{dt} = 2x + 3y

given that $x(0) = 4$ and $y(0) = 2$ .

[7]

Question 11

Find the particular solution of the decoupled system

\frac{dx}{dt}=x, \\ \frac{dy}{dt}=3y,

with $x(0)=4$ , $y(0)=2$ .

[4]

Question 12

The solution involves solving a system of first-order linear differential equations with constant coefficients. This requires finding the eigenvalues and eigenvectors of the coefficient matrix and applying initial conditions to find the constants.

Solve the system of differential equations

\begin{aligned} \frac{dx}{dt} &= 2y \\ \frac{dy}{dt} &= -x + 5y \end{aligned}

with initial conditions $x(0) = 4$ and $y(0) = 2$ .

[7]

Question Type 2: Finding the particular solution to linear coupled differential equations with some initial condition and real and distinct eigenvalues Exercises