Differential Equations are extremely helpful to solve complex mathematical problems in almost every domain of Engineering, Science and Mathematics. If you are an Engineer, you will be integrating and differentiating hundreds of equations throughout you career, because these equations have a hidden answer to a really complex problem. Mathematicians and Researchers like Laplace, Fourier, Hilbert etc., have developed such equations to make our life easier.
Maxwell has given equations of Electromagnetic Field Theory, which help us to find the Magnetic Field or Electric Field easily by simple integration/differentiation. Various Transforms from Time Domain to Frequency Domain or vice versa in Engineering is only possible because of Differential Equations. In Mechanical/Civil Engineering, people use such equations for solving complex fluid dynamics problems, and finding the right balance of weights and measures to build stuff like a Cantilever Truss, for example. Partial Derivatives are used to find maxima and minima of functions with more than 2 dependent variables, while there are differential equations to find the complexity of a contour in case of Complex Numbers (Eg. Cauchy integral Theorem).
These are just few of examples which I could quote. There are tons and tons of applications of Differential Equations in Engineering. And it is really exciting when you get a hidden answer by solving those complex differential equations, really.