Question

these are doubts related to example 16 and 17 of differential equations , the final ans of 16 that is integral cosv=integral 1/x dx has been given as sinv=logx +logc (log is taken with the constant) whereas in example 17 integral 2e^​vdv= -integral dy/y is given as 2e^v= -logy +c (log is no taken with c) both the questions has terms involving log then what is the difference?

Answer 1

Dear student,

$\int \frac{1}{x}dx=\log \left(x\right)\hspace{0.17em}+\hspace{0.17em}c$
But, the value of constant can be anything.

We can say that there is a constant v whose value is equal to log(c). In that case we can write:

$\int \frac{1}{x}dx=\log \left(x\right)+\hspace{0.17em}v=\log \left(x\right)\hspace{0.17em}+\hspace{0.17em}\log \left(c\right)\hspace{0.17em}$

I hope this clarifies your doubt and answers your question.

Regards