Question

Solving initial value problems Solve the following initial value problems. $$y^{\prime}(t)=1+e^{t}, y(0)=4$$

Short answer

Answer: y(t) = t + e^t + 3

Step-by-step solution

Integrate the Given Equation

To find the general solution of the ODE, we need to perform integration. We are given $$y^{\prime}(t)=1+e^{t}.$$ Integrate both sides with respect to t: $$\int y^{\prime}(t) \, dt = \int (1+e^{t}) \, dt.$$

Perform Integration

When integrating both sides, we get: $$y(t) = \int (1+e^{t}) \, dt.$$ Integration yields: $$y(t) = t + e^{t} + C,$$ where C is the integration constant.

Apply the Initial Condition

Now we apply the initial condition y(0) = 4 to find the value of the constant C: $$4 = y(0) = 0 + e^{0} + C.$$

Solve for the Constant C

Simplify the equation to find the value of C: $$4 = 1 + C,$$ The value of the constant C is: $$C = 3.$$

Write the Final Solution

Now that we have the value of C, we can write the final solution to the given initial value problem: $$y(t) = t + e^{t} + 3.$$

Key concepts

Integrating Ordinary Differential Equations

Understanding how to integrate ordinary differential equations (ODEs) is a fundamental skill for solving various problems in mathematics, physics, and engineering. An ODE is an equation that involves a function and its derivatives. The process of integration is aimed at finding a function that satisfies the given differential equation. The general solution typically contains one or more integration constants, which represent the infinite possibilities of solutions corresponding to different initial conditions.

For instance, when presented with an equation like \( y'(t) = 1 + e^t \), the integration process involves finding an antiderivative of the right-hand side with respect to the variable \(t\). This process is very much like finding the area under a curve or reversing the process of differentiation to find the original function. A proper understanding of integral calculus is essential to perform this step accurately.

Applying Initial Conditions

After determining the general solution, the next critical step involves applying initial conditions to find the particular solution to the ODE. Initial conditions are given values of the function and/or its derivatives at specific points, which help to determine the specific instance of the general solution that fits the problem at hand.

Using the initial condition \(y(0) = 4\), we plug the value of \(t = 0\) into the general solution to find the specific constant value that makes the equation true. This process of substituting the initial conditions into the general solution helps to 'pin down' a single, unique solution among the infinite possibilities. It's much like fitting the missing piece into a puzzle – once the correct position is found, the overall picture becomes clear.

Integration Constants

In the context of integrating ODEs, the integration constants represent the 'unknown' parts of the general solution. These constants arise because when you take the integral of a derivative, there's an infinite number of functions that could have resulted in the same derivative. These constants are the key to transitioning from a general to a particular solution.

The value of the integration constant \(C\) in our example is determined by applying the initial condition. Thus, solving \(4 = 0 + e^{0} + C\) yields \(C = 3\). The role of the integration constant is pivotal because the correct assignment of this value directly impacts the accuracy and relevance of the final solution within the given constraints.

Exponential Functions

Exponential functions, like \(e^t\), are frequently encountered when solving differential equations. They have unique properties that make them particularly important, such as being the only functions whose derivative is proportional to the function itself.

An understanding of the behavior of exponential functions—how they grow rapidly, their role in modeling continuous growth or decay processes—is imperative. When integrating an ODE that includes an exponential term, we essentially reverse-engineer to find the original exponential function that, when differentiated, would yield the given rate of growth or decay. Recognizing the integral of \(e^t\) as simply \(e^t\) is crucial, as it allows for swift progress through the integration steps.