To solve the problem where we have the differential equation \( \frac{dy}{dx} = 2x - 3 \) and the initial condition \( y = 3 \) when \( x = 0 \), we need to find the function \( y \) in terms of \( x \). Let's go through the steps in detail. ### Step 1: Integrate the Differential Equation The first step is to integrate the right-hand side of the equation to find \( y \). The equation we have is: \[ \frac{dy}{dx} = 2x - 3 \] To find \( y \), we integrate both sides with respect to \( x \): \[ y = \int (2x - 3) \, dx \] ### Step 2: Perform the Integration Now, we perform the integration: 1. The integral of \( 2x \) is \( x^2 \). 2. The integral of \( -3 \) is \( -3x \). Putting it all together, we have: \[ y = x^2 - 3x + C \] where \( C \) is the constant of integration that we need to determine using the initial condition. ### Step 3: Apply the Initial Condition We know from the problem that when \( x = 0 \), \( y = 3 \). We can use this information to find \( C \): \[ 3 = (0)^2 - 3(0) + C \] This simplifies to: \[ 3 = C \] ### Step 4: Write the Final Equation Now that we have determined \( C \), we can substitute it back into our equation for \( y \): \[ y = x^2 - 3x + 3 \] ### Conclusion: Identify the Correct Option Now, let's compare our final equation \( y = x^2 - 3x + 3 \) with the provided options: - A. \( 2x^2 - 3x \) - B. \( x^2 - 3x \) - C. \( x^2 - 3x - 3 \) - D. \( x^2 - 3x + 3 \) The correct option is **D**: \( x^2 - 3x + 3 \). ### Explanation of Incorrect Options - **Option A: \( 2x^2 - 3x \)**: This option is incorrect because it does not match the integrated function. The coefficient of \( x^2 \) should be 1, not 2. - **Option B: \( x^2 - 3x \)**: This option is missing the constant term \( +3 \). It does not satisfy the initial condition \( y = 3 \) when \( x = 0 \). - **Option C: \( x^2 - 3x - 3 \)**: This option has the wrong constant term. It would yield \( y = 0 \) when \( x = 0 \), which does not satisfy the initial condition. ### Revision Summary - To find \( y \) from \( \frac{dy}{dx} \), integrate the right-hand side. - Use the initial condition to find the constant of integration. - The final function is \( y = x^2 - 3x + 3 \). - Always check each option against the derived function and initial conditions to confirm correctness.