Finding the solution of a differential equation has become very mechanical and I don't know why are we finding an equation that solves the differential equation.
Being inverse to integration, and integration is saying, "the sum of", then what is this solution of a differential equation leading to in integration?
A differential equation involves unknown functions and derivatives. You are asking why we solve differential equations, I will give you some reasons: Suppose you want to find the area of some curve $$ 0 \le x \le 1, ~ y=g(x), ~\text{where $g(x)$ is continuosly differentiable.} $$ For this purpose you need to solve integral $ \int_{0}^{1}g(x)dx$, $~$ by Fundamental theorem of calculus, integral will be $$ f(1)-f(0), ~ \text{where $f$ satisfies}~ f'(x)=g(x). $$ Which requires solution of differential equation $f'(x)=g(x). $ $$ \text{Suppose some ball is thrown from a height} ~H~ \text{with speed $u$, then height of ball satisfies:} $$ $$ \frac{d^{2}h}{dt^{2}}=-g,~ \text{where $g$ is constant acceleration due to gravity.} $$ Then at any time $t$, if you want to find the height of ball and velocity of ball, you need to solve differential equations. As a concluding remark, I would say differential equation comes naturally while modelling many physical phenomenona.
Part of the reason Calculus gained attention in its early days after the dust has settled between mathematicians, is because it has enabled mathematics to take a greater roll in describing the real world, for example, the acceleration and velocity of objects in a slick way. This is because Calculus deals with change, unlike other branches of Mathematics like Geometry. The universe is not constant, it is constantly changing. Being able to describe change is very important for the many scientific studies. To reap real rewards, the scientists need to describe the change and be able to extract more information about it, like predicting the behavior of a phenomena under different inputs, or at different points in time. After all, we can't keep observing nature forever and record data. We got to jump to generalizations to form solid facts. Functions provide an ideal tool to know much about the behavior of a special kind of relations based on parameters (variables).
In the real world, you are rarely given $f(x)$!. If you are lucky you can collect data, you build a model and you interpret the model. Most of the science you could get from Calculus begins to be useful only when you determine the function (even a close approximation may do). How can you do that? This is where the science of Differential Equations kicks in. It helps you extract the function(s) at the heart of the models you are studying from the observed change hence allowing you to utilize the abstract functions to generate more and more knowledge about the phenomena at hand. Needles to say, Differential Equations enabled mathematicians to explore more about the fabric that ties different branches of mathematics.
It is a shame that many authors don't include concrete applications of the mathematical concepts they write books about!
This is my humble understanding of it all.
The constituent equations governing a geometrical, physical, or sociological context are found by thinking geometrically or physically, etc., about the setup under question. The result of this thinking are equations that contain as variables time, coordinates of points, curvatures, speeds, accelerations, and more. Some of these quantities are derivatives of others, e.g., momentary speed is the derivative of the locality $x$ of a moving point with respect to time $t$.
It follows that a found equation $$\Psi(x,y, t, v,\ldots)=0\tag{1}$$ between the involved variables is often a differential equation, e.g., $y''-3y'-y=\cos t$. Thinking about a real system we want to know whether there are equilibrium points, or, under which circumstances will the actual solution $t\mapsto y(t)$ evolve to $\infty$, and how fast. In order to answer such questions we have to be able to solve the equation $(1)$. Unfortunately this is only seldom possible to do in finitary mathematical formulas. These are the cases that you have learned to handle "mechanically".
