IB Mathematics Applications & Interpretation Question Type 3: Approximating Values for Coupled System Using Euler's Method for a Given Step Length Exercises

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: $$\frac{\text{d}x}{\text{d}t}=t+x,\frac{\text{d}y}{\text{d}t}=y-t$$ given the initial conditions $x=0$ and $y=2$ at $t=0$.

Using Euler’s method with step length $h=0.5$, approximate $x$ and $y$ at $t=2.0$ for the system: $$\frac{dx}{dt}=x-y+2t,\frac{dy}{dt}=3x+y^2,$$ with initial values $t=1$, $x=2$, $y=0$.

Using Euler’s method with step length $h=0.2$, approximate $x$ and $y$ at $t=0.8$ for the system of differential equations: $$\frac{dx}{dt}=x^2+y^2,\frac{dy}{dt}=2x-y+t^2,$$ with initial values $x=4$ and $y=2$ at $t=0$.

Using Euler’s method with step length $h=0.5$, approximate $x$ and $y$ at $t=2.0$ for the system: $$\frac{dx}{dt}=x^2+y^2,\frac{dy}{dt}=2x-y+t^2$$ with initial values $t=0,x=4,y=2$.

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

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

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

Using Euler’s method with step length $h=0.5$, approximate $x$ and $y$ at $t=1.0$ for the system: $$\frac{dx}{dt}=tx+y,\frac{dy}{dt}=x+ty$$ with initial values $t=0,x=1,y=2$.

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: $$\frac{dx}{dt}=x^2+y^2,\frac{dy}{dt}=2x-y+t^2,$$ with initial values $x(0)=4$ and $y(0)=2$.

Using Euler’s method with step length $h=0.2$, approximate $x$ and $y$ at $t=1.6$ for the system: $$\frac{dx}{dt}=2x+3y,\frac{dy}{dt}=x-y+t,$$ with initial values $t=1$, $x=0$, $y=1$.

Using Euler’s method with step length $h=0.1$, approximate $x$ and $y$ at $t=0.3$ for the system: $$\frac{dx}{dt}=x-y+t,\frac{dy}{dt}=x+y^2,$$ with initial values $t=0$, $x=1$, $y=0$.

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