Solve each differential equation using the given boundary...

Solve each differential equation using the given boundary condition. (Hint: Rewrite each differential equation in the form $g(y) d y=f(x) d x$ and integrate both sides.
$$
\frac{d y}{d x}=e^{x}, y=4 \text { when } x=0
$$

Key Concepts

Separation of Variables

This method involves rearranging a differential equation so that all terms involving one variable (and its differential) are on one side of the equation and all terms involving the other variable are on the opposite side. It is a standard technique for solving first-order differential equations by separately integrating each side with respect to its own variable.

Integration

After separating the variables, the next step is to integrate both sides of the equation. Integration provides a means to accumulate the effects of the differential changes and ultimately reach a general solution involving an arbitrary constant.

Constant of Integration

When an indefinite integral is computed, an arbitrary constant is added to represent the family of solutions. This constant is later determined by applying a given initial or boundary condition, ensuring that the solution fits the specific constraints of the problem.

Initial Conditions

An initial or boundary condition provides a specific value for the function at a certain point, which is used to determine the constant of integration. This process allows one to find a unique solution from the general solution of a differential equation.

Exponential Function

Exponential functions are a common type of function in differential equations. Their properties, such as the fact that the derivative of an exponential function is proportional to the function itself, often simplify the integration process and lead to straightforward solutions.

Transcript

Please give Ace some feedback