problem in bernoulli equation when n=1

[pooya1072]

hi friends .... sorry if this is a very elementary question.
In bernoulli equation we have:
1 – y’+p(x)y=g(x)y
2 – (dy/y)+[p(x)-g(x)]dx=0
3 - ln y = integral {[g(x)-p(x)]dx }
4 – y=c * Exp(integral{ [g(x)-p(x)]dx )}
A) I want to know how 3th line derive to 4th line? b) Why c Multiple by exp?
thanks

 

[HallsofIvy]

hi friends .... sorry if this is a very elementary question.
In bernoulli equation we have:
1 – y’+p(x)y=g(x)y
2 – (dy/y)+[p(x)-g(x)]dx=0
3 - ln y = integral {[g(x)-p(x)]dx }
4 – y=c * Exp(integral{ [g(x)-p(x)]dx )}
A) I want to know how 3th line derive to 4th line? b) Why c Multiple by exp?
thanks

That is an "elementary question" but one worth asking! First, you should have, in line 3, "integral {[g(x)- p(x)]dx}+ c" where c is the "constant of integration".

Now take the exponential of both sides:
\(\displaystyle e^{ln(y)}= e^{\int (g(x)- p(x))dx+ c}= e^{\int (g(x)-p(x)dx}e^c\)
where I have used the property of exponentials, \(\displaystyle a^{b+ c}= a^ba^c\).
Of course \(\displaystyle e^{ln(y)}= y\) so that
\(\displaystyle y= Ce^{\int (g(x)- p(x))dx}\)
where \(\displaystyle C= e^c\).

 

[pooya1072]

That is an "elementary question" but one worth asking! First, you should have, in line 3, "integral {[g(x)- p(x)]dx}+ c" where c is the "constant of integration".

Now take the exponential of both sides:
\(\displaystyle e^{ln(y)}= e^{\int (g(x)- p(x))dx+ c}= e^{\int (g(x)-p(x)dx}e^c\)
where I have used the property of exponentials, \(\displaystyle a^{b+ c}= a^ba^c\).
Of course \(\displaystyle e^{ln(y)}= y\) so that
\(\displaystyle y= Ce^{\int (g(x)- p(x))dx}\)
where \(\displaystyle C= e^c\).

Thanks HallsofIvy
That’s right . but I’m confusing about c , as I know c is appear after integral operation, but here it came before integral operation .for example :
\(\displaystyle \int (X)dx =(0.5*x^2) +Constant \)
That constant Create after integral operation , so why in :
\(\displaystyle \int {(g(x)- p(x))dx}+ constant \)
Constant placed here,whereas it must placed in for example this form:
\(\displaystyle \int {(g(x)- p(x))dx} = ............. + constant \)

 

[HallsofIvy]

Thanks HallsofIvy
That’s right . but I’m confusing about c , as I know c is appear after integral operation, but here it came before integral operation .for example :
\(\displaystyle \int (X)dx =(0.5*x^2) +Constant \)
That constant Create after integral operation , so why in :
\(\displaystyle \int {(g(x)- p(x))dx}+ constant \)
Constant placed here,whereas it must placed in for example this form:
\(\displaystyle \int {(g(x)- p(x))dx} = ............. + constant \)

It is not true that it must be placed anyhere!

If you had \(\displaystyle \int (g(x)- p(x))dx+ constant= .......\), do you see that that is the
same as \(\displaystyle \int (g(x)- p(x))dx= .........- constant\)?

And do you see that, since "constant" is an arbitrary number, so is "-constant" and it really doesn't matter which you write?

 

[pooya1072]

ok .... thank you very much.