Verify the existence and uniqueness of solutions for the following differential equation: \(\frac{dy}{dx} = 3y + 4x\)

Step 1 ：Step 1: For the existence and uniqueness of the solution of the differential equation, we use the Existence and Uniqueness Theorem, which states that if the function \(f(x,y)\) and its partial derivative \(\frac{\partial f}{\partial y}\) are continuous in a rectangle containing the point \((x_0, y_0)\), then there is a unique solution to the initial value problem.

Step 2 ：Step 2: Here, \(f(x,y) = 3y + 4x\). The partial derivative of \(f\) with respect to \(y\) is \(\frac{\partial f}{\partial y} = 3\), which is continuous for all \((x, y)\).

Step 3 ：Step 3: Also, \(f(x,y) = 3y + 4x\) is continuous for all \((x, y)\).

Step 4 ：Step 4: Therefore, by the Existence and Uniqueness Theorem, there exists a unique solution to the given differential equation.