Problem 56 - Calculus

Question

In Problems \(55-58,\) give an example of: An expression for \(f(x)\) such that the differential equation \(d y / d x=f(x)+x y-\cos x\) is separable.

Step-by-Step Solution

Answer

Choose \( f(x) = \cos(x) \) to make the equation separable as \( \frac{dy}{dx} = xy \).

Step 1: Understand Separable Differential Equations

A separable differential equation can be written in the form \( g(y) \, dy = h(x) \, dx \), where we can separate variables and integrate both sides independently.

Step 2: Analyze the Given Differential Equation

The given differential equation is \( \frac{dy}{dx} = f(x) + xy - \cos(x) \). For this to be separable, we need to express \( y' \) as a product of a function of \( x \) and a function of \( y \).

Step 3: Determine the Form of \( f(x) \)

To make the equation \( \frac{dy}{dx} = f(x) + xy - \cos(x) \) separable, \( f(x) \) must be chosen such that the entire expression becomes a function that allows separation. A straightforward choice is \( f(x) = \cos(x) \) because subtracting \( \cos(x) \) will eliminate it from the equation.

Step 4: Rework the Differential Equation

Substitute \( f(x) = \cos(x) \) into the equation: \( \frac{dy}{dx} = \cos(x) + xy - \cos(x) = xy \). This simplifies to \( \frac{dy}{dx} = xy \), a separable form.

Step 5: Perform Variable Separation

The equation \( \frac{dy}{dx} = xy \) can be rewritten as \( \frac{1}{y} \, dy = x \, dx \), where the variables are separated, making integration straightforward.

Key Concepts

Variable Separation

Variable separation is a fundamental technique in solving separable differential equations. The main goal of this method is to reorganize a differential equation so that all terms involving the variable 'y' are on one side, and all terms involving the variable 'x' are on the other side. This allows each side of the equation to be integrated independently.

• To achieve variable separation, you manipulate the original differential equation into a form like this: \( g(y) \, dy = h(x) \, dx \).
• This process often requires clever manipulation or intuition to rewrite the equation.
• Once separated, it prepares the differential equation for the next step: integration.

The elegance of this method lies in its simplicity; once the variables are clearly separated, solving the differential equation becomes much more manageable through integration. It's a critical step to transform complicated differential equations into solvable integrals.

Integration

After successfully separating the variables, the next hurdle is using integration to solve the differential equation. Integration is the process of finding the integral of a function, which is essentially the reverse operation of differentiation.
To solve a separable differential equation, you integrate both sides:

• The left side of the separated equation, often involving 'y', requires integrating with respect to 'y'.
• The right side, often involving 'x', requires integrating with respect to 'x'.
• The antiderivatives you find will ultimately help describe the relationship between 'x' and 'y'.

In our current context, integrating the equation \(\frac{1}{y} \, dy = x \, dx \) results in:

• \( \ln |y| = \frac{1}{2}x^2 + C \) as antiderivatives, where \(C\) is a constant of integration.

This process of solving the integrals gives us a function that describes how 'y' changes with 'x'. Remember, integration, especially definite integration, brings out constant terms that describe specific solution scenarios in these differential systems.

Function of x and y

In the context of separable differential equations, understanding the roles and interactions between the function of 'x' and the function of 'y' is crucial. These functions dictate how the solution of the differential equation manifests.

• 'x' typically represents the independent variable.
• 'y' represents the dependent variable, as its solution depends on 'x'.
• The function \(f(x)\) is selected or derived to simplify the differential equation into a separable form.

In our example, choosing \(f(x) = \cos(x)\) allowed substituting into the equation such that it yielded a product \(xy\), something easily separable. The function of \(x\) and \(y\) must be aligned correctly for separation and integration to work seamlessly.
A successful separation leads to an equation like \(\frac{1}{y} \, dy = x \, dx\), paving the way to find integrals that solve for 'y' in terms of 'x'. Recognizing these functions' roles is key to unlocking the solution within separable differential equations.