dx=cos5x make your answer correct and complete solution not direct solution the identify what equation in that question make sure correct your answer.

To solve the differential equation \( \frac{dx}{dt} = \cos(5x) \), let's go through the problem step by step, ensuring we understand the type of equation and how to solve it.

### Step 1: Identify the Type of Differential Equation
The given equation is:
\[
\frac{dx}{dt} = \cos(5x)
\]
This is a **separable** first-order differential equation because the right-hand side depends only on \( x \) and not on \( t \). This means we can separate the variables \( x \) and \( t \) on different sides of the equation.

### Step 2: Separate the Variables
We want to isolate \( dx \) on one side and \( dt \) on the other side. To do this, divide both sides by \( \cos(5x) \) and multiply both sides by \( dt \):
\[
\frac{1}{\cos(5x)} \, dx = dt
\]

### Step 3: Integrate Both Sides
Now that we have separated the variables, integrate both sides with respect to their respective variables:
\[
\int \frac{1}{\cos(5x)} \, dx = \int dt
\]
The integral on the left side involves \( \frac{1}{\cos(5x)} \), which can be recognized as the secant function:
\[
\int \sec(5x) \, dx = \int dt
\]
To solve \( \int \sec(5x) \, dx \), we use a known antiderivative:
\[
\int \sec(5x) \, dx = \frac{1}{5} \ln |\sec(5x) + \tan(5x)| + C_1
\]
Where \( C_1 \) is the constant of integration. So, the integral of the left side is:
\[
\frac{1}{5} \ln |\sec(5x) + \tan(5x)| = t + C_2
\]
Where \( C_2 \) is another constant of integration.

### Step 4: Combine and Simplify the Equation
Combine the constant terms:
\[
\frac{1}{5} \ln |\sec(5x) + \tan(5x)| = t + C
\]
Where \( C = C_2 - C_1 \) is the general constant of integration.

### Step 5: Solve for \( x \) (Optional)
If you need to explicitly solve for \( x \), you can proceed as follows:
\[
\ln |\sec(5x) + \tan(5x)| = 5(t + C)
\]
Exponentiate both sides:
\[
|\sec(5x) + \tan(5x)| = e^{5(t + C)} = e^{5t} \cdot e^{5C} = A e^{5t}
\]
Where \( A = e^{5C} \) is a new constant. Thus:
\[
\sec(5x) + \tan(5x) = \pm A e^{5t}
\]
Solving explicitly for \( x \) can be complex, so often this is left in implicit form unless initial conditions are given.

### Conclusion
The solution to the differential equation is:
\[
\frac{1}{5} \ln |\sec(5x) + \tan(5x)| = t + C
\]
or equivalently,
\[
\sec(5x) + \tan(5x) = \pm A e^{5t}
\]

### Relative Questions:
1. How do you solve a non-separable differential equation?
2. What are the properties of the secant and tangent functions?
3. How would you apply initial conditions to find the particular solution?
4. What techniques can be used to solve other types of first-order differential equations?
5. What is the role of the constant of integration in differential equations?

### Tip:
Always check if a differential equation can be separated, as this greatly simplifies the process of finding a solution.

Learn how to solve the differential equation dx/dt = cos(5x) step-by-step with detailed explanations. This problem involves separable differential equations, integration techniques, and exponential functions. Suitable for undergraduate-level mathematics.

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