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Question 1

Numerical solutions to differential equations: Use of Euler's method for a system of two first-order differential equations.

Use Euler’s method with a step length of $h = 0.2$ to approximate the values of $x$ and $y$ at $t = 0.6$ for the system of differential equations:

\frac{\text{d}x}{\text{d}t} = t + x, \quad \frac{\text{d}y}{\text{d}t} = y - t

given the initial conditions $x = 0$ and $y = 2$ at $t = 0$ .

[5]

Question 2

Using Euler’s method with step length $h = 0.5$ , approximate $x$ and $y$ at $t = 2.0$ for the system:

\frac{dx}{dt} = x - y + 2t, \quad \frac{dy}{dt} = 3x + y^2,

with initial values $t = 1$ , $x = 2$ , $y = 0$ .

[6]

Question 3

Using Euler’s method with step length $h=0.2$ , approximate $x$ and $y$ at $t=0.8$ for the system of differential equations:

\frac{dx}{dt} = x^2 + y^2, \quad \frac{dy}{dt} = 2x - y + t^2,

with initial values $x=4$ and $y=2$ at $t=0$ .

[6]

Question 4

Using Euler’s method with step length $h = 0.5$ , approximate $x$ and $y$ at $t = 2.0$ for the system:

\frac{dx}{dt} = x^2 + y^2, \quad \frac{dy}{dt} = 2x - y + t^2

with initial values $t = 0, x = 4, y = 2$ .

[6]

Question 5

Using Euler’s method with step length $h=0.1$ , approximate $x$ and $y$ at $t=0.2$ for the system: $\frac{dx}{dt} = x^2 - y, \quad \frac{dy}{dt} = y^2 - x + t,$ with initial values $t=0$ , $x=0$ , $y=1$ .

[4]

Question 6

Using Euler’s method with step length $h=0.1$ , approximate $x$ and $y$ at $t=0.2$ for the system:

\frac{dx}{dt} = xy, \quad \frac{dy}{dt} = x - y

with initial values $x=1, y=1$ at $t=0$ .

[4]

Question 7

Using Euler’s method with step length $h=0.1$ , approximate $x$ and $y$ at $t=0.2$ for the system:

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

with initial values $t=0$ , $x=2$ , $y=1$ .

[4]

Question 8

Numerical solutions to differential equations: Euler's method for systems.

Using Euler’s method with step length $h = 0.5$ , approximate $x$ and $y$ at $t = 1.0$ for the system:

\frac{dx}{dt} = tx + y, \quad \frac{dy}{dt} = x + ty

with initial values $t = 0, x = 1, y = 2$ .

[6]

Question 9

Numerical solutions to differential equations: Euler's method for systems of first-order differential equations.

Using Euler’s method with step length $h = 0.1$ , approximate $x$ and $y$ at $t = 0.4$ for the following system of differential equations:

\frac{dx}{dt} = x^2 + y^2, \quad \frac{dy}{dt} = 2x - y + t^2,

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

[6]

Question 10

This question assesses the ability to use Euler's method for a system of coupled first-order differential equations to approximate values at a specific time step.

Using Euler’s method with step length $h=0.2$ , approximate $x$ and $y$ at $t=1.6$ for the system:

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

with initial values $t=1$ , $x=0$ , $y=1$ .

[5]

Question 11

Calculus: Euler's method for systems of differential equations.

Using Euler’s method with step length $h=0.1$ , approximate $x$ and $y$ at $t=0.3$ for the system:

\frac{dx}{dt} = x - y + t, \quad \frac{dy}{dt} = x + y^2,

with initial values $t=0$ , $x=1$ , $y=0$ .

[5]

Question 12

Using Euler’s method with step length $h=0.2$ , approximate $x$ and $y$ at $t=0.4$ for the system: $\frac{dx}{dt} = x + t^2 y, \quad \frac{dy}{dt} = t x^2 - y$ with initial values $t=0, x=1, y=1$ .

[5]

Question Type 3: Approximating values for coupled system using Euler's method for a given step length Exercises