Who knows how to solve this? Some one who knows physics maybe?
Answer:
Let -2b = A.
dy = {e^ (At) } * dt.
Integrating this we get y = A * e^ (At) + some constant.
y = -2b * e^ (-2bt) + some constant
Meaning of integral
Consider integral x dx.
x is multiplied by dx
Now think of a line whose length is x and thickness ‘dx’, a small amount of x
In the line of length x ,draw a perpendicular line of length dx.
Now x dx is the AREA of the rectangle
formed by the side x and dx.
The integral of x dx means we stake such rectangles side by side and find the area when all dx’s add to the length x.
dx============================...
dx============================...
dx============================...
dx============================...
In the above figure the thickness between the pair of lines is dx and length is x.
There are gaps between the pairs of lines. Now stake them one over the so that there is no gap in-between the pairs of lines.
Keep doing this till the thickness becomes x
Now the area of the figure is x^2 . The length is x and breadth is x.
In the final answer we have no dx.
Another example.
Integral x^2 dx.
Take a piece of square paper.
Let the sides be x.
The area of the square paper is x^2.
The thickness of the paper is dx.
Therefore x^2 * dx denotes the volume of the paper.
Stake such papers one above the other so that the thickness of the papers all put together comes to x.
The volume of the paper is now x^3.
Therefore integral x^2 dx is x^3.
Now I hope you know where the dx has gone.
All the dx’s have contributed to the side of the paper bundle.
That is the meaning of integral.
(dt) in the first equation tells you that you are integrating with relation to time (t)...hence there is no dt in the answer.
The "dt" indicates a differential of the variable "t". Think of a differential as being a bunch of tiny parts of something; when you intergrate, you get the whole thing, hence you no longer have that bunch of little parts. That's what an intergral is really; summing of a lots tiny things that are varying in some way (time, position, etc).
That's about as non-technical as I can make it.
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