Question Type 4: Approximating values for coupled system using Euler's method with fixed iteration and step length using technology Exercises

Use Euler's method with step size $h = 0.1$ to approximate the solution at $t = 0.5$ for the system of differential equations: $\frac{\mathrm{d}x}{\mathrm{d}t} = x^2 + y^2$ $\frac{\mathrm{d}y}{\mathrm{d}t} = 2x - y + t^2$ given the initial conditions $x(0) = 4$ and $y(0) = 2$.

Using Euler’s method with a step size of $h=0.1$, approximate the values of $x(0.3)$ and $y(0.3)$ for the system of coupled differential equations:

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

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

Use Euler’s method with step length $h=0.25$ to find $x(0.5)$ and $y(0.5)$ for the system of differential equations: $\frac{\text{d}x}{\text{d}t}=x^2+y^2$ $\frac{\text{d}y}{\text{d}t}=2x-y+t^2$ given the initial conditions $x(0)=4$ and $y(0)=2$.

Approximate $x(1.0)$ and $y(1.0)$ using Euler’s method with step size $h = 0.5$ for the system of differential equations $\frac{dx}{dt} = x^2 + y^2$ and $\frac{dy}{dt} = 2x - y + t^2$, given the initial conditions $x(0) = 4$ and $y(0) = 2$.

Use Euler's method with two steps of size $h = 1.0$ to approximate $x(2.0)$ and $y(2.0)$ for the system of differential equations $\frac{dx}{dt} = x^2 + y^2, \quad \frac{dy}{dt} = 2x - y + t^2$ with initial conditions $x(0) = 4$ and $y(0) = 2$.

Estimate $x(0.75)$ and $y(0.75)$ by Euler’s method with step size $h = 0.25$ for the system of differential equations

$\frac{\text{d}x}{\text{d}t} = x^2+y^2$ $\frac{\text{d}y}{\text{d}t} = 2x-y+t^2$

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

With step size $h = 0.5$, apply Euler’s method to approximate $x(1.5)$ and $y(1.5)$ for the system of differential equations: $\frac{dx}{dt} = x^2 + y^2, \quad \frac{dy}{dt} = 2x - y + t^2$ given the initial conditions $x(0) = 4$ and $y(0) = 2$.

Starting from $x(0)=4, y(0)=2$, use Euler’s method with $h=0.2$ to approximate $x(0.4)$ and $y(0.4)$ for $\frac{dx}{dt}=x^2+y^2$ and $\frac{dy}{dt}=2x-y+t^2$.

Using a step size of $h=0.25$, apply Euler’s method to approximate $x(1.0)$ and $y(1.0)$ for the system of differential equations $\frac{dx}{dt} = x^2 + y^2$ $\frac{dy}{dt} = 2x - y + t^2$ with the initial conditions $x(0) = 4$ and $y(0) = 2$.

Apply Euler’s method with step size $h = 0.1$ to estimate $x(0.4)$ and $y(0.4)$ for the system $\frac{dx}{dt} = x^2 + y^2$, $\frac{dy}{dt} = 2x - y + t^2$, starting at $(0, 4, 2)$. Give your final answers to three significant figures.

For the system of differential equations

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

with initial conditions $x(0) = 4$ and $y(0) = 2$, use Euler’s method with step length $h = 0.1$ to approximate the values of $x(0.2)$ and $y(0.2)$.