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

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

Use Euler’s method to approximate $y(0.6)$ for $\frac{dy}{dx}=x^2+y^2$ with the initial condition $y(0)=1$ and a step size of $h=0.2$.

Given $\frac{\text{d}y}{\text{d}x} = x^2 + y^2$ and $y(0) = 1$, use Euler’s method with step size $h = 0.5$ to approximate $y(1)$. State the number of steps taken in the approximation.

Derive the Euler iteration formula for this differential equation and calculate the values of $y_1, y_2,$ and $y_3$ using a step size of $h = 0.1$. Give your final answers to three significant figures.

Approximate $y(0.2)$ using Euler’s method with the differential equation $\frac{dy}{dx} = x^2 + y^2$, initial condition $y(0) = 1$, and a step size of $h = 0.05$. Give your answer to four decimal places.

Approximate $y(0.4)$ using Euler’s method for the differential equation $\frac{dy}{dx}=x^2+y^2$ with the initial condition $y(0)=1$, using a step size $h=0.1$.

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 $h = 0.1$.

Use Euler’s method to approximate $y(1)$ for the differential equation $\frac{dy}{dx} = x + y$, given the initial condition $y(0) = 1$ and a step size of $h = 0.2$.