Differential Equations: Numerical Methods

Given the differential equation $\frac{dY}{dX} = Y$ and the initial condition $Y(0) = 1$, use Euler's method with a step size of 1 to approximate $Y(1)$.

Given the differential equation $\frac{dY}{dX} = Y$ and the initial condition $Y(0) = 1$, use Euler's method with a step size of 0.5 to approximate $Y(1)$.

Use Euler's method with a step size of 0.02 to approximate $y(0.08)$ for the given differential equation with initial condition $x_0 = 0, y_0 = 1$.

Estimate the value $Y(4)$ for the initial value problem: $y' - y = 0$ with $Y(0) = 1$ using Euler's method and a step size of 1.

Given that $\frac{dy}{dx} = 3x + y$ and the initial condition $y(0) = -1$, approximate $y(0.2)$ with a step size of $0.04$ using Euler's Method.

Using the Improved Euler's method, solve for the approximate value of $y$ at $x = 1.3$ for the differential equation $y' = x \cdot y$ with the initial condition $y(1) = 1$ and a step size $h = 0.1$.

Using the Euler and improved Euler techniques, approximate the values of $y$ given the initial conditions and a step size.

Given $\frac{dy}{dx} = f(x, y)$ with $x_0 = 3, y_0 = 2$, step size $h = 0.1$, approximate $y$ when $x = 3.1$ using the Euler method.

Using Euler's method with step size $h = 0.1$, approximate $y$ when $x = 1.1$ given that $x_0 = 1, y_0 = 1$ and $f(x, y) = x + 3 + \sin(y)$.

Using the improved Euler's method, approximate $y$ when $x = 1.2$ given that $x_0 = 1, y_0 = 1$, $h = 0.1$, and $f(x, y) = x + 3 + \, \sin(y)$.

Using the improved Euler method, with initial conditions $x_0 = 0$ and $y_0 = 1$, and a step size $h = 0.5$, calculate the value of $y$ for $x = 0.5$.

Use Runge-Kutta with step size $H = 0.1$ to estimate $y(0.2)$ in the initial value problem $y' = t^2 + y^2$ and $y(0) = 1$.

Using Euler's method, approximate the solution to the differential equation $y' = f(x, y)$ starting from the point $(x_0, y_0)$ and proceeding with steps $x_1, x_2, \ldots, x_n$.

Given a differential equation $y' = x \cdot y$, approximate the y-value when $x = 1.3$ with an initial condition of $y(1) = 1$ and increment $h = 0.1$.

Given $y' = y$ with an initial condition $y(0) = 1$ and step size $h = 0.01$, use the Euler method to approximate the solution of the differential equation.

Given $y' = y - x^2$ with initial condition $y(0) = 0$ and step size $h = 0.1$, use the Euler method to approximate the solution of the differential equation.