1. Which of the following is a solution to the differential equation
ylny + xy’ = 0 for x > 0 ?
|
A. xlny = 1 |
B. xylny = 1 |
C. (lny)2 = 2 |
|
D. –y(lny)(lnx) = 1 |
E. lny + (x2/2)y = 1 |
Solution: The answer is A
To solve the differential equation,
First solve for y’, as follows;
Notice that this is a first order separable differential equation whose solution is given by separating the variables and their corresponding differentials on each side of the equation and integrating both sides. Hence,
By substitution, that is, let
, then 
and thus, 
Substituting this result in equation (1) and integrating the right side of equation (1) we obtain the following equation:
Where k is a constant. Thus, xlny = 1 is a solution to the differential equation ylny + xy’ = 0 for x > 0.