Answer to Problem 31

Write (4.3) as



Equation 20

We shall solve for by "separating variables". This technique may be used only when cross-multiplying enables us to put all expressions involving one variable (and its differential) on the left side of the equation and all expressions involving the other variable (and its differential) on the right side, as can be done here. It makes no difference on which sides constants appear. Multiplying both sides of the equation above by , we obtain



Equation 21

Then from



Equation 22

we have



Equation 23

and arbitrary constants.

Then is also an arbitrary constant, and we get



Equation 24

where is an arbitrary constant.