Solve the given differential equation by using an appropriate substitution. The DE is of the form dy/dx= f(Ax + By + C), which is given in (5) of Section 2.5. dy/dx = tan^(2)(x + y)

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.

This solution demonstrates how to solve the differential equation dy/dx = tan^2(x + y) using substitution. The process includes applying trigonometric identities and separating variables. Ideal for undergraduate students studying differential equations and trigonometric methods.

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