Sure! I'll tackle **Question 2** step-by-step, as requested: --- ### **2. Find the general solution for this differential equation.** \[ \frac{dy}{dt} = \sec^2 (t) \] where \( y \) is the dependent variable and \( t \) is the independent variable. --- #### **Step 1: Recognize the Type of Equation** This is a **first-order ordinary differential equation (ODE)** and is *separable*. We can write: \[ \frac{dy}{dt} = \sec^2 (t) \] To solve, integrate both sides with respect to \( t \): \[ \int \frac{dy}{dt} dt = \int \sec^2 (t) dt \] **Explanation:** We're using the property that the derivative of \( \tan(t) \) is \( \sec^2(t) \), and integrating both sides with respect to the independent variable \( t \). --- #### **Step 2: Integrate Both Sides** \[ y(t) = \int \sec^2 (t) dt = \tan(t) + C \] where \( C \) is the constant of integration. **Explanation:** This follows directly from the standard integral: \[ \int \sec^2 (t) dt = \tan(t) + C \] --- ### **Final Answer:** \[ \boxed{y(t) = \tan(t) + C} \] where \( C \) is an arbitrary constant.

Sure! Let's analyze the question step-by-step: --- ### **Step 1: Recognize the type of differential equation** The given differential equation is: \[ \frac{dy}{dt} = \sec^2 (t) \] **Explanation:** This is a first-order differential equation where the derivative of \( y \) with respect to \( t \) is given explicitly. Since the right side depends only on \( t \), it is directly integrable with respect to \( t \). --- ### **Step 2: Integrate to find the general solution** Integrate both sides with respect to \( t \): \[ y(t) = \int \sec^2(t) dt + C \] Using the standard integral: \[ \int \sec^2(t) dt = \tan(t) + C' \] where \( C' \) is an arbitrary constant (which can be renamed as \( C \)). **Explanation:** The integral of \( \sec^2(t) \) is a well-known result, \( \tan(t) + C \). --- ### **Final answer:** \[ \boxed{y(t) = \tan(t) + C} \] where \( C \) is an arbitrary constant of integration.