If where , then the value of at is
Explanation for the correct option.
Step 1. Find the general equation of .
It is given that , so .
The corresponding equation is: .
Let the general solution be given by the equation .
Now find the value of and .
It is given that at , the value of is . So substitute for and for in the equation .
Now differentiate equation , with respect to .
It is given that at , the value of is . So substitute for and for in the equation .
Now subtract equation from equation .
Now substitute for in equation .
So in the general equation substitute for and for to get the solution of the differential equation.
Step 2. Find the value of at .
In the equation substitute for and find the value of using properties of logarithm.
So, the value of at is .
Hence, the correct option is D.