How to find f(x) if df(x)dx=f(x)? I know cex is a solution, but how does one find it and how to prove it is the complete solution?

Name: How to find f(x) if df(x)dx=f(x)? I know cex is a solution, but how does one find it and how to prove it is the complete solution?
Uploaded: 2022-02-23T17:22:57+00:00
Description: How to find f(x) if df(x)dx=f(x)? I know cex is a solution, but how does one find it and how to prove it is the complete solution?

Question

scomparve5j · Accepted Answer

To find it, you can use separation of variables. If you&#039;re accustomed to recognizing that f′f=log⁡(|f|)′, you can integrate directly as Chandru1 noted. Otherwise, you can use the substitution u=f.Even without knowing what the solutions are, note that if f and g are solutions and g is nonzero, then (fg)′=f′g−g′fg2=, so fg is constant, which means f=cg for some constant c. This shows that the multiples of one nonzero solution form the complete solution. In general, a first order homogeneous linear ODE has a one dimensional solution space.

limacarp4 · Accepted Answer

Assume f:R→R and the equation if valid for all x∈R.Trying to divide by f or trying to talk in terms of logf would need more justification (what if f is zero in some interval?) and has the potential to lose some solutions (though I am pretty sure there will be some theory there which will justify the correctness of the method).A simpler way, which avoids taking cases:f′(x)=f(x)⇔e−xf′(x)−e−xf(x)=0⇔(e−xf(x))′=0e−xf(x)=c

alenahelenash · Accepted Answer

Just integrate. You have∫f′(x)f(x)dx=dxtherefore you have log⁡f(x)=x+C, So f(x)=ex+C=ex.eC=kex.

How to find f(x) if \frac{df(x)}{dx}=f(x)? I know ce^{x} is

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