91Ó°ÊÓ

The substitution method is a powerful tool used in solving differential equations, making it simpler by transforming complex equations into more manageable forms. In this particular problem, the need for substitution arises when we notice the composite argument \(x+y\) within the tangent squared term, \(\tan^2(x+y)\).

By introducing a substitution such as \(u = x + y\), we convert the variable expression into a single variable form. This substitution makes the equation easier to manipulate and solve. Differentiating \(u = x + y\) with respect to \(x\) gives us \(\frac{du}{dx} = 1 + \frac{dy}{dx}\), providing an equation that relates \(\frac{dy}{dx}\) to \(\frac{du}{dx}\).

After substitution, the original equation \(\frac{dy}{dx} = \tan^2(u)\) gets modified to \(\frac{du}{dx} - 1 = \tan^2(u)\), which simplifies further into \(\frac{du}{dx} = 1 + \tan^2(u)\). This demonstrates how a clever substitution can transform complex expressions, revealing a simpler path to the solution.

Trigonometric identities are fundamental relationships involving trigonometric functions that allow for the simplification of expressions. In this exercise, the identity \(1 + \tan^2(u) = \sec^2(u)\) plays a crucial role.

This identity is derived from the Pythagorean identities and enables us to rewrite expressions involving \(\tan^2(u)\) in a simplified form. It essentially tells us that the sum of 1 and the square of the tangent is equivalent to the square of the secant function.

By applying this identity, the equation \(\frac{du}{dx} = 1 + \tan^2(u)\) is transformed into \(\frac{du}{dx} = \sec^2(u)\). Trigonometric identities like this one are key to breaking down and solving complex trigonometric equations. Mastering them provides an invaluable shortcut in a variety of mathematical problems, especially those involving trigonometric expressions.

Separable differential equations are a type of differential equations that can be written in a form where all terms involving one variable can be separated from all terms involving another variable. This makes them relatively straightforward to integrate.

In the problem at hand, once the substitution and trigonometric identity have been applied, the equation becomes \( \frac{du}{dx} = \sec^2(u)\). This equation is naturally separable. We can re-arrange it to get \( du = \sec^2(u) \, dx\).

The separable form allows us to integrate both sides independently. Integrating the left side \( \int du \) gives us \( u \), and the integral of \( \int \sec^2(u) \, dx \) results in \( \tan(u) + C \). The result is a relationship \( u = \tan(u) + C \) which, through substitution back, describes a relationship between the original variables \(x\) and \(y\).

This method highlights the power and simplicity of separable differential equations, offering a structured approach to solving what might initially seem like complex differential problems.