Calculus of a Single Variable, Chapter 6, 6.3, Section 6.3, Problem 5

The general solution of a differential equation in a form of can be evaluated using direct integration. The derivative of y denoted as y' can be written as (dy)/(dx) then y'= f(x) can be expressed as (dy)/(dx)= f(x)
For the problem (dr)/(ds)=0.75r , we may apply variable separable differential equation in which we set it up as f(y) dy= f(x) dx .
Then,(dr)/(ds)=0.75r can be rearrange into (dr)/r=0.75 ds .

Applying direct integration on both sides:
int (dr)/r= int 0.75 ds .
For the left side, we apply the basic integration formula for logarithm: int (du)/u = ln|u|+C
int (dr) /r = ln|r|
For the right side, we may apply the basic integration property: int c*f(x)dx= c int f(x) dx .
int 0.75 ds=0.75int ds .
Then the indefinite integral will be:
0.75int ds= 0.75s+C
Combining the results for the general solution of differential equation:
ln|r|=0.75s+C
r= Ce^(0.75s)