Question Type 1: Approximating values for a given step length using Euler's method Exercises

Approximate $y(0.1)$ using Euler’s method with $h=0.05$ for $\frac{\text{d}y}{\text{d}x} = x^2 + y^2$, given $y(0)=1$.

Use Euler’s method with step size $h = 0.1$ to approximate $y(0.3)$ for $\frac{dy}{dx} = x^2 + y^2$, given the initial condition $y(0) = 1$.

Use Euler’s method with step size $h = 0.2$ to approximate $y(1.0)$ for the differential equation $\frac{\text{d}y}{\text{d}x} = x^2 + y^2$, given the initial condition $y(0) = 1$.

Use Euler’s method with step size $h=0.1$ to approximate the value of $y$ at $x=0.4$ for the differential equation $\frac{dy}{dx}=x^2+y^2$ with initial condition $y(0)=1$.

Use Euler’s method with step length $h = 0.05$ to approximate $y(0.2)$ for the differential equation $\frac{dy}{dx} = x^2 + y^2$, given the initial condition $y(0) = 1$.

Use Euler’s method with step size $h = 0.25$ to estimate $y(0.75)$ for the differential equation $\frac{dy}{dx} = x^2 + y^2$, given the initial condition $y(0) = 1$.

Apply Euler’s method with step length $h = 0.2$ to approximate $y(0.6)$ for the differential equation $\frac{dy}{dx} = x^2 + y^2$, given the initial condition $y(0) = 1$.

Approximate $y(2.0)$ using Euler’s method with a step size of $h = 0.5$ for the differential equation $\frac{dy}{dx} = x^2 + y^2$, given the initial condition $y(0) = 1$.

Use Euler’s method with step size $h=0.25$ to estimate $y(1.0)$ for $\frac{dy}{dx}=x^2+y^2$, given $y(0)=1$.

Use Euler’s method with $h = 1.0$ to estimate $y(4)$ for $\frac{dy}{dx} = x^2 + y^2$, given the initial condition $y(0) = 1$.

Apply Euler’s method with step size $h=0.1$ to estimate $y(0.5)$ for the differential equation $\frac{dy}{dx}=x^2+y^2$ with the initial condition $y(0)=1$.