There has always been some petty difficulty in grasping the intuitive idea of a differential equation, maybe due to the endeavor required in deriving the implication of such or due to the analysis required in arriving at the solution (which can't be easily visualized for a beginner). Thus, in this text I will attempt to explain it in as conceptual an approach can be (but I won't cover technicalities like how to arrive at a solution given this kind of differential equation because any standard textbook can do that).
To begin with, I shall review the basic meaning of a differential:
d/dx - RATE OF CHANGE
d2/dx2 - RATE OF CHANGE OF THE RATE OF CHANGE
Naturally, we aren't very much concerned with the higher derivatives if we are still new to the topic because they represent the rate of change of the rate of change of the rate of change of the rate of change, ... of some characteristic of a system. And if it existed, i.e. the higher order differentials were non-zero, then we would have an extremely unstable system in our hands (can you imagine an object changing acceleration every now and then? -3rd order- and that rate of change changing? -4th order- ) which then becomes a seldom concern of engineers.
Why do we need differential equations? Because in reality, we are not only dealing with stationary things. Our world, our environment is always changing. Movement (velocity) and the rate of change of that movement (acceleration), centripetal acceleration of revolving objects such as celestial bodies, rate of combustion, rate of growth, rate of dissipation, rate of diffusion, rate of heating and cooling, etc. Differential equations are essential and they are here to stay (at least for the modelling process, later I shall explain why phasors work in simplifying analysis of electrical circuits, and how transforms with different kernels help make differential equation solutions algebraic).
The solution of a differential equation is the function that will satisfy the differential equation. But what exactly IS a solution to a differential equation. It is simply trying to express the nature of a system in more basic dimensions (the relationship of the displacement with time INSTEAD of the relationship of the velocity/acceleration with time or the relationship of the springs displacement in time instead of the relationship of its damping coefficient, mass, velocity and acceleration with time). We solve differential equations because we want a function/machine that will tell us immediately what the basic dimension y is given an independent variable x.
The solution of a differential equation always takes a general form composed of 2 parts, the general solution (possessing arbitrary constants) and the particular solution. But why is it that way? What does a general solution mean? Why is there a particular part? Well, the general solution is the solution that can predict the nature of the differential equation given zero initial conditions. In control systems, it is referred to as the natural response. Without an input/initial condition, the general solution suffices. BUT, when there is an input/initial condition to the differential equation (the system being modelled is not at its set point/there is an initial disturbance/the system is not initially in equilibrium), the general solution is not valid and we need a particular solution. Thus, the particular solution is the solution that when added to the general solution will make the entire function valid for a given input/initial condition. The particular solution is commonly called the forced response.
Phasors are an essential tool in simplifying ac electric circuit analysis. But how exactly do phasors do this for engineers? The nature of inductors and capacitors are mathematically described by differentials, with loops forming differential equations. How could the analysis become a simple "vector" calculation? By assuming that the solution to the differential equation, i.e. the voltage/current in any branch of the circuit will ALWAYS BE SINUSOIDAL IN NATURE. By using exp(j*2*pi*f*t) as the immediate solution to ALL differential equations formed in any loop, the variation will only be in the amplitude and the phase - 2 values represented as x and y - a vector! This immediately implies that phasor analysis is good ONLY for sinusoidal inputs (which is usually ok since a lot can be concluded from the response of a circuit from a single sinusoidal frequency input).
Now we finally deal with transforms. There is nothing mystifying about transforms, you are simply convolving a function with another function (usually standardized and is complicatedly termed the "kernel" of the transform). (Remember that convolution is like multiplying the frequency content of 2 signals.) If the standard function you convolved with is exponential, we call the convolution a Laplace transform. If the exponential's exponent is complex, we call the convolution a Fourier transform. If we note that an exponential is simply a cosine and complex sine, then we are projecting the function to sines and cosines when we are convolving with an exponential. No wonder you get the spectrum of a signal when you use the Fourier transform since the frequency components of the signal are projected to sinusoids. (Also note that the laplace transform's counterpart is the z transform when we are dealing with discrete-time discrete valued signals)
So how does the laplace transform help with solving differential equations? I still don't have the intuitive train of thought that leads to the answer but somehow, when you take the laplace transform of a differential/a rate of change, the term becomes algebraic! Then, you do some little simple algebra (like transposing, adding, subtracting...), take the inverse transform and you end up with the solution. There is a catch though, you have to know the initial conditions. If you don't know the initial conditions or you assume that there is none, you won't arrive at the correct solution (unless your assumption is by chance correct).