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

For the system $\frac{dx}{dt}=x^2+y^2$, $\frac{dy}{dt}=2x - y + t^2$ with $x(0)=4$, $y(0)=2$, use Euler’s method with step length $h=0.1$ to approximate $x(0.2)$ and $y(0.2)$.

Using Euler’s method with $h=0.1$, approximate $x(0.3)$ and $y(0.3)$ for $\frac{dx}{dt}=x^2+y^2$, $\frac{dy}{dt}=2x-y+t^2$, $x(0)=4$, $y(0)=2$.

Use Euler’s method with step length $h=0.25$ to find $x(0.5)$ and $y(0.5)$ for $dx/dt=x^2+y^2$, $dy/dt=2x-y+t^2$, $x(0)=4$, $y(0)=2$.

Apply Euler’s method with step size $h=0.1$ to estimate $x(0.4)$ and $y(0.4)$ for the system $dx/dt=x^2+y^2$, $dy/dt=2x-y+t^2$, starting at $(0,4,2)$.

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 $dx/dt=x^2+y^2$, $dy/dt=2x-y+t^2$.

Estimate $x(0.75)$ and $y(0.75)$ by Euler’s method with $h=0.25$ for $dx/dt=x^2+y^2$, $dy/dt=2x-y+t^2$, $x(0)=4$, $y(0)=2$.

Approximate $x(1.0)$ and $y(1.0)$ using Euler’s method with $h=0.5$ for $dx/dt=x^2+y^2$, $dy/dt=2x-y+t^2$, given $x(0)=4$, $y(0)=2$.

Use Euler’s method with $h=0.1$ to approximate the solution at $t=0.5$ for $dx/dt=x^2+y^2$, $dy/dt=2x-y+t^2$, given $x(0)=4$, $y(0)=2$.

Use a single step of size $h=1.0$ to approximate $x(2.0)$ and $y(2.0)$ for $dx/dt=x^2+y^2$, $dy/dt=2x-y+t^2$, with $x(0)=4$, $y(0)=2$.

Using $h=0.25$, apply Euler’s method to approximate $x(1.0)$ and $y(1.0)$ for the system $dx/dt=x^2+y^2$, $dy/dt=2x-y+t^2$, with $x(0)=4$, $y(0)=2$.

With $h=0.5$, apply Euler’s method to approximate $x(1.5)$ and $y(1.5)$ for the system $dx/dt=x^2+y^2$, $dy/dt=2x-y+t^2$, $x(0)=4$, $y(0)=2$.