Why do we need differential equations.

Question

1) why are equations of the form,

dy/dx = f(x,y) i.e dy/dx = f(x) g(y), where y is y(x) i.e f(x). necessary? what is the need of differential equations.

2) Suppose in the above equation we have,f(x) as √x and g(y) = y+1. but y is y(x) i.e h(x) so g(y)=g[y(x)] right? so what we substitute g[h(x)] in y+1 ? how?

3)why do we integrate it after showing the equality.i mean, for e.g here we have to show that rate of change of y=h(x) is equal to the product of f(x) and g(y). what is the reason for integrating.

Answer 1

Hi Rahul A.,

Differential equations are one way of expressing the relationships between two variables (or more!). As such, they may be the simplest representation of the essential causes and effects, as it were, of many quantities in the real world.

It's a little hard to follow the jumble of your question, but I think you're asking about the equation

dy/dx = ((x)^0.5)*(y+1) ?

That's an exemplar of a separable differential equation. You can separate each variable, and its differential, to the same side of the equation, and solve easily. So:

dy/(y+1) = x^0.5 * dx

ln (y+1) + C = (x^1.5)/1.5

Since that ln term on the left is added to a constant, both sides of the equation may be exponentiated:

c(y+1) = e^(x^1.5/1.5) where ln(C) = c.

[You don't care about that (C vs. c), since the constant is arbitrary, until you have to solve for it to meet a specified initial condition.]

That's a perfectly acceptable form of stating the solution, but you can probably figure out how to extract just y = f(x), right?

Now, as to why you would like to express one variable as a function of only the other variable (if you can: sometimes differential equations are only soluble numerically, or are best left as parametric relationships of a "dummy" variable): that's because you are greatly familiar with the forms of some basic functions: lines, parabolas, ln(x), among many others. Therefore, you will grasp the underlying relationship between the co-behavior of the two variables, when one is such a function of the other.

For example, I could tell you the value of the acceleration (2nd derivative of position vs. t) of a particular object. But that alone doesn't help you predict where it will be any particular time, does it? Whereas, an expression for position vs. time, typically a parabola, that tells you a lot.

--Cheers, --Mr. d.