Problem 18 - Calculus Early Transcendentals

Question

Solve the differential equation and then use a graphing utility to generate five integral curves for the equation. $$ (\cos y) y^{\prime}=\cos x $$

Step-by-Step Solution

Answer

The solution is \(\sin y = \sin x + C\), with integral curves for various \(C\) values.

Step 1: Identify the Type of Differential Equation

The given differential equation is \((\cos y) y' = \cos x\). This is a first-order, separable differential equation because it can be rewritten in the form \(f(y)y' = g(x)\).

Step 2: Separate Variables

Rearrange the equation to separate the variables. Divide both sides by \(\cos y\) to get: \[ y' = \frac{\cos x}{\cos y} \] This can be written as: \[ \cos y \, dy = \cos x \, dx \] The variables are now separated.

Step 3: Integrate Both Sides

Integrate both sides of the equation to solve for \(y\). The left side becomes: \[ \int \cos y \, dy = \sin y + C_1 \] The right side becomes: \[ \int \cos x \, dx = \sin x + C_2 \] Combine the integration constants into one constant \(C\): \[ \sin y = \sin x + C \]

Step 4: Solve for y

Since the equation \(\sin y = \sin x + C\) doesn’t lend itself easily to solving explicitly for \(y\), the solution will be implicit in terms of the relationship between \(x\) and \(y\).

Step 5: Generate Integral Curves

Use a graphing utility to plot the curves represented by\[ \sin y = \sin x + C \] Select different values for \(C\) (e.g., \(C = -1, -0.5, 0, 0.5, 1\)) to plot five integral curves. Each curve will represent a different solution based on the initial condition.

Key Concepts

Separable Differential Equations

Separable differential equations are a special type of differential equations where the variables can be separated onto different sides of the equation. This makes these equations particularly straightforward to solve.
In a separable differential equation, the general form can be represented as:

• \( f(y) y' = g(x) \)

To separate variables, one manipulates the equation to isolate all terms involving \( y \) on one side and all terms involving \( x \) on the other. In this way, we transform the equation into something resembling:

• \( f(y) dy = g(x) dx \)

Once in this form, the solution process becomes easier because you can integrate both sides independently. This step leads to solutions expressed implicitly, or sometimes explicitly, in terms of \( y \) and \( x \).
This structured separation and integration facilitate many real-world calculations very efficiently.

Integral Curves

Integral curves are graphical representations of solutions to differential equations. They can offer visual insights into the behavior of the differential solutions. Each integral curve corresponds to a particular solution that satisfies the given differential equation under specific initial conditions.
In the context of the equation \(\sin y = \sin x + C\), these curves are generated for various constant values \( C \). Each curve illustrates how as \( C \) varies, the solution to the equation behaves differently. You can think of each integral curve as a path that a point following the differential equation might trace out over time.
By using a graphing utility to visualize these curves, it becomes easier to understand complex relationships and dynamics described by differential equations. It's particularly useful because many differential equations can't be easily solved or interpreted without the aid of such visual tools.

First-Order Differential Equations

First-order differential equations involve derivatives of the first degree. They have the general form \( \frac{dy}{dx} = f(x, y) \). These equations are fundamental and often appear in natural and physical sciences because they describe how changes in one variable relate directly to changes in another.
There are different methods to tackle first-order differential equations, but when they are separable (like in our exercise), it allows us to use variable separation to find the solution. First-order differential equations can take in a wide variety of forms, but if you can convert them into a separable form, it simplifies the solution process significantly, allowing you to focus on the pure integration part which you typically already know how to handle.

Variable Separation

Variable separation is a technique used to solve certain types of differential equations, particularly separable ones. This method involves rearranging the equation in such a manner that all terms involving one variable are on one side and all terms involving the other variable are on the opposite side.
For example, starting with this equation:

• \( (\cos y) y' = \cos x \)

You would rearrange it by dividing both sides by \(\cos y\), resulting in:

• \( \cos y \, dy = \cos x \, dx \)

This isolating step is crucial because it allows each side of the equation to be integrated separately, leading directly to a solution to the differential equation. This process of separating the dependent and independent variables simplifies solving many otherwise challenging problems and is a foundational tool in the study of differential equations.