We are given the differential equation:

$\frac{dy}{dx} = \tan^2(x + y)$

This differential equation is of the form $\frac{dy}{dx} = f(Ax + By + C)$, where $f(u) = \tan^2(u)$ and $A = B = 1$, $C = 0$.

Step 1: Use the substitution $u = x + y$

We can apply the substitution $u = x + y$ to simplify the equation.

Thus, we have:

$\frac{du}{dx} = \frac{d}{dx}(x + y) = 1 + \frac{dy}{dx}$

Using the given differential equation $\frac{dy}{dx} = \tan^2(x + y) = \tan^2(u)$, the equation becomes:

$\frac{du}{dx} = 1 + \tan^2(u)$

Step 2: Simplify the equation

Recall that $1 + \tan^2(u) = \sec^2(u)$, a standard trigonometric identity. So, the equation becomes:

$\frac{du}{dx} = \sec^2(u)$

Step 3: Separate variables

We can now separate variables to solve the differential equation:

$\frac{du}{\sec^2(u)} = dx$

Since $\frac{1}{\sec^2(u)} = \cos^2(u)$, this becomes:

$\cos^2(u) du = dx$

Step 4: Integrate both sides

Now, integrate both sides:

$\int \cos^2(u) \, du = \int dx$

We use the identity $\cos^2(u) = \frac{1 + \cos(2u)}{2}$ to rewrite the integral:

$\int \frac{1 + \cos(2u)}{2} \, du = \int dx$

The left-hand side can be integrated as follows:

$\frac{1}{2} \int (1 + \cos(2u)) \, du = \frac{1}{2} \left( u + \frac{\sin(2u)}{2} \right) = \frac{u}{2} + \frac{\sin(2u)}{4}$

The right-hand side integrates to $x + C$, where $C$ is the constant of integration. So, we have:

$\frac{u}{2} + \frac{\sin(2u)}{4} = x + C$

Step 5: Substitute back $u = x + y$

Finally, substitute back $u = x + y$:

$\frac{x + y}{2} + \frac{\sin(2(x + y))}{4} = x + C$

This is the implicit solution to the differential equation.

Would you like more details or have any questions?

Here are some related questions to deepen understanding:

1. How would the solution change if the equation were $\frac{dy}{dx} = \tan^2(2x + y)$?
2. What are other common trigonometric identities that can simplify integrals?
3. How does the substitution method work for higher-order differential equations?
4. What are some physical applications of this type of differential equation?
5. How do you solve differential equations of the form $\frac{dy}{dx} = e^{x + y}$?

Tip: For trigonometric integrals, memorizing key identities like $\sec^2(x) = 1 + \tan^2(x)$ can help simplify and solve problems faster.