Approximating with Eulers 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 $f(x, y) = x + 3 + \sin(y)$.

Euler's method is a popular numerical tool used to approximate solutions to ordinary differential equations (ODEs). At its core, Euler's method is based on the concept of solving an ODE by converting it into a finite difference problem and iteratively applying tangent line approximations to generate a sequence of approximate values. The method is particularly useful when analytical solutions to differential equations are difficult or impossible to find. By choosing an appropriate step size, one can manage the trade-off between computational workload and approximation accuracy, as a smaller step size generally leads to more accurate approximations but requires more computations. In this problem, you are asked to use Euler's method with a specific step size to approximate a value of the dependent variable given initial conditions. It's crucial to start from the initial condition and use the differential equation to compute the slope at each step, which then gives a new approximation of the dependent variable for a slightly updated value of the independent variable. This iterative process is repeated until the desired value of the independent variable is reached. Understanding Euler's method provides insight into the broader field of numerical methods and their applicability in solving differential equations that model real-world phenomena across science and engineering disciplines.