IB Mathematics Applications & Interpretation 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:

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

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 $$\frac{dx}{dt}=x^2+y^2,\frac{dy}{dt}=2x-y+t^2$$ 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: $$\frac{dx}{dt}=x^2+y^2,\frac{dy}{dt}=2x-y+t^2$$ 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 $y(0.2)$.