Question Type 2: Approximating values for a given interation amount and step length using technology Exercises

Derive the Euler iteration formula for $\frac{dy}{dx}=x^2+y^2$ and compute $y_1,y_2,y_3$ with $y(0)=1$, $h=0.1$.

Use Euler’s method to approximate $y(1)$ for the differential equation $\frac{dy}{dx}=x^2+y^2$ with initial condition $y(0)=1$ and step size $h=0.1$.

With $h=0.5$, use Euler’s method to approximate $y(1)$ for $\frac{dy}{dx}=x^2+y^2$, $y(0)=1$. Also state how many steps are taken.

Use Euler’s method to approximate $y(2)$ for $\frac{dy}{dx}=x^2+y^2$ with $y(0)=1$ and $h=0.1$.

Approximate $y(1)$ using Euler’s method with $\frac{dy}{dx}=x^2+y^2$, $y(0)=1$ and step size $h=0.05$.

Use Euler’s method to approximate $y(3)$ for $\frac{dy}{dx}=x^2+y^2$ with $y(0)=1$ and a larger step size $h=0.2$.

Approximate $y(1.5)$ using Euler’s method for $\frac{dy}{dx}=x^2+y^2$ with $y(0)=1$, step size $h=0.1$.

Use Euler’s method to estimate $y(2.5)$ for $\frac{dy}{dx}=x^2+y^2$, $y(0)=1$, with $h=0.1$.

Approximate $y(3)$ again with $h=0.1$ using Euler’s method for $\frac{dy}{dx}=x^2+y^2$, $y(0)=1$, and compare with the result for $h=0.2$. Compute the absolute difference.

Approximate $y(4)$ using Euler’s method for $\frac{dy}{dx}=x^2+y^2$, $y(0)=1$, with step size $h=0.1$.

Approximate $y(4)$ again with Euler’s method but now use $h=0.05$ for $\frac{dy}{dx}=x^2+y^2$, $y(0)=1$.