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.