Solve the Initial Value Problem 𝑑𝑦𝑑𝑥 = cos 𝑥 , 𝑦(0) = 1.

To solve the initial value problem, we need to integrate the function and apply the initial condition. Here are the steps:

1. We start with the differential equation dy/dx = cos(x).

2. To find y(x), we need to integrate both sides with respect to x. The integral of dy is y, and the integral of cos(x) dx is sin(x) + C, where C is the constant of integration. So, we have y = sin(x) + C.

3. Now, we apply the initial condition y(0) = 1. Substituting these values into the equation gives us 1 = sin(0) + C. Since sin(0) = 0, we find that C = 1.

4. Therefore, the solution to the initial value problem dy/dx = cos(x), y(0) = 1 is y = sin(x) + 1.