Question

For the following problems, find the general solution to the differential equation.\(y^{\prime}=\sin x e^{\cos x}\)

Short answer

The general solution is \( y = -e^{\cos x} + C \).

Step-by-step solution

Identify the Type of Differential Equation

The given differential equation is a first-order ordinary differential equation (ODE): \( y' = \sin x \cdot e^{\cos x} \). We need to find its general solution.

Integrate with Respect to x

To solve the differential equation, we integrate both sides with respect to \( x \):\[ y = \int \sin x \cdot e^{\cos x} \, dx. \]

Use Substitution to Simplify Integration

Let \( u = \cos x \), then \( du = -\sin x \, dx \) or \(-du = \sin x \, dx \). Substitute to get:\[ y = \int e^{u} (-du) = -\int e^{u} \, du. \]

Integrate

The integral of \( e^u \, du \) is straightforward:\[ -\int e^u \, du = -e^u + C, \] where \( C \) is the constant of integration.

Substitute Back to Original Variable

Replace \( u \) back with \( \cos x \) to express the solution in terms of \( x \):\[ y = -e^{\cos x} + C. \]

Write the General Solution

Finally, we have the general solution of the differential equation:\[ y = -e^{\cos x} + C. \]

Key concepts

First-order ODE

A first-order ordinary differential equation (ODE) is an equation that involves the first derivative of an unknown function and the function itself. It is called "first-order" because it involves the first, and only the first derivative. In the example given, we have a first-order ODE in the form of:\[ y' = f(x, y) \]Here, the function depends on the variable \(x\) and the function of \(y\), but often it's expressed in terms of just \(x\), as seen in our exercise:\[ y' = \sin x \cdot e^{\cos x}.\]The primary goal when dealing with a first-order ODE is to find a general solution or expression for \(y\) as a function of \(x\). This is achieved by integrating the ODE with respect to \(x\), which leads us to finding a general formula for \(y\). By doing so, we translate the problem from dealing with derivatives to dealing with integrals, a shift that often simplifies the problem substantially.

Integration by Substitution

Integration by substitution is a powerful technique used to simplify integrals, especially when dealing with composite functions. It's a fundamental method wherein we change the variable of integration to make the integral easier to solve. This method can be compared to reversing the chain rule in differentiation.In the context of the original differential equation given:\[ y = \int \sin x \cdot e^{\cos x} \, dx,\]substitution helps us tackle the complexity inherent in the integral. We begin by identifying a part of the integrand that can be replaced with a simpler variable, \(u\). Here, we let:\[ u = \cos x \ du = -\sin x \, dx \]This transforms the integral into a much simpler form:\[ y = \int e^{u} (-du) = -\int e^{u} \, du\]Through substitution, the integral becomes straightforward to evaluate, leading us to answers more easily than would have been possible with the original variables. Integration by substitution is especially helpful when dealing with functions that seem too complex or convoluted initially.

Constant of Integration

When solving differential equations, particularly indefinite integrals, it is crucial to include the constant of integration. This constant, often denoted as \(C\), represents an arbitrary constant that arises because the process of integration can only determine functions up to an additive constant.In our solution:\[ y = -e^{\cos x} + C\]The \(C\) is necessary because when differentiating, any constant term vanishes, and thus does not appear in the derivative. When we introduce \(C\), we're considering all potential solutions that differ by a constant amount, which collectively represent the 'general solution'.The constant of integration is particularly important when we have initial conditions or boundary values, as these allow us to solve for \(C\) and find a unique solution to the differential equation. Without \(C\), any solution obtained would essentially be incomplete, unable to account for all possible functions that satisfy the differential equation.