# General solution to a simple ODE

## Main Question or Discussion Point

Whenever I am stuck I usually manage by sitting down and working on the problem and eventuall finding the solution, this one is bothering me too much and I don't have any class untill friday so no hope of finding out before then unless I ask here.

Q: Find a general solution to the diff.eq:

d[f(x)]/dx = bf(x). Given f(0) = 1 and f'(0) = 3 define constants and find a solution for f(x)

Attempts:
Stuck, used 2th order ODEs so much this thing confuses me.

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This is solved simply through seperation of variables:

$$\frac{df}{dx}=bf$$

Therefore

$$\frac{df}{f}=bdx$$

Performing indefinite integration over this gives you:

$$ln(f)=bx+c$$

With c being arbitrary constant of integration
And explicitly f is given by

$$f(x)=ae^{bx}$$

Where I chose to rewrite the arbitrary constant e^c as a.

Determining a and b from your additional conditions is a simple algebric exercise.

This is solved simply through seperation of variables:

$$\frac{df}{dx}=bf$$

Therefore

$$\frac{df}{f}=bdx$$

Performing indefinite integration over this gives you:

$$ln(f)=bx+c$$

With c being arbitrary constant of integration
And explicitly f is given by

$$f(x)=ae^{bx}$$

Where I chose to rewrite the arbitrary constant e^c as a.

Determining a and b from your additional conditions is a simple algebric exercise.
Thank you!