Question

In Exercises \(1-10,\) find the general solution to the exact differential equation. $$\frac{d y}{d t}=(\cos t) e^{\sin t}$$

Short answer

The general solution to the differential equation \(\frac{d y}{d t} = \cos(t) e^{\sin t}\) is \(y = e^{\sin(t)} + C\).

Step-by-step solution

Identify the function to be integrated

The function to be integrated is \(\cos(t) e^{\sin t}\), which is the right hand side of the given differential equation.

Integrate the function

To find the solution of the given ODE, we integrate the function \(\cos(t) e^{\sin t}\) with respect to \(t\). The integral is \(\int \cos(t) e^{\sin t} dt\). To solve this, we use the method of substitution, letting \(u=\sin(t)\). The differential of \(u\) is then \(du=\cos(t) dt\). Substituting into the integral gives us: \(\int e^u du\).

Compute the integral and find the general solution

The integral \(\int e^u du\) is \(e^u + C\), where \(C\) is the constant of integration. Substituting back \(u=\sin(t)\) gives the solution: \(e^{\sin(t)} + C\).

Key concepts

Integrating Factors

When faced with solving differential equations, integrating factors can be a crucial tool. These are functions used to multiply the original differential equation in order to transform it into an exact differential equation, which is often easier to solve. An exact differential equation has a form that allows it to be integrated directly to find a solution. To apply an integrating factor, one must have a solid understanding of differential equations and an intuition for selecting the right type of function that will simplify the equation to its integral form. It's essentially like finding a key that unlocks the equation, making it more manageable.

In the exercise provided, the equation is already given in a form that can be directly integrated without the use of an integrating factor. However, it's beneficial for students to recognize that not all differential equations will be this straightforward, and integrating factors are a powerful method to know for those tougher cases.

Method of Substitution

The method of substitution is like a magic trick in calculus. It involves changing the variables in an integral to simplify it, making the expression easier to manage and solve. It's a transformative step that can turn a complex-looking integral into one that is much more familiar. Substitution is usually used when an integral contains a function and its derivative. In the given exercise, we identify \(u = \sin(t)\) as a substitution because the derivative of \(\sin(t)\), which is \(\cos(t)\), also appears in the integral.

By replacing \(dt\) with \(\frac{du}{\cos(t)}\), the integral becomes \(\int e^u du\), a much simpler integral to compute. After performing the integration, we substitute back the original variables to get the general solution. This method is widely used not only in solving differential equations but also in a variety of calculus problems.

Constant of Integration

In the realm of calculus, the constant of integration represents the indefinite nature of antiderivatives. Every time you integrate a function, there is an infinite number of possible functions that could have been the original derivative. These possible functions differ by a constant value, and hence, we include the constant of integration, denoted as \(C\), to represent these possibilities. It's the 'C' that ensures we acknowledge all possible antecedent functions.

For example, in our exercise, after integrating \(e^u\) with respect to \(u\), we correctly include the constant of integration in our final answer, \(e^{\sin(t)} + C\). This \(C\) is critical because without it, we would be ignoring a whole family of potential solutions. The constant of integration is a fundamental concept that secures the completeness of the solution to an indefinite integral or differential equation.