when to introduce the constant C in differential equations ?

[King]

when to introduce the constant C in differential equations ?

Exactly after the integration and then you simplify , or first you integrate ,simplify and then introduce C ?

Because in some answers I have solutions with K simplified .. like :

e^x = x + Kx

And now I am a bit confused :S

 

[michaelempeigne]

insufficient information.

 

[Red Jacket]

To understand why we add the constant C you must know some basics.Consider the following
y=sin x --------(1)
Differentiating with respect to x we get
dy/dx =cos x
Now consider the equation
y=sin x +10 --------(2)
Differentiating with respect to x we get
dy/dx =cos x +0 =cos x

Notice that we get the same derivative for both the functions 1 & 2 because the derivative of a constant like 10 is 0.Since integration is the inverse process of differentiation,we have to get the original function from the derivative.As we have no way to find out whether the original function had a constant by just looking at it's derivative,we add an arbitrary constant 'C'.C can be any possible value including 0 as in function 1.

As to introducing C in differential equations you can add it in the very final step.

I am quite sure that ?(e^x)dx = e^x + C .Do clarify this.

Hope this helps.