Question 02 (b)
Solve the following ODE
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
This equation is exact (you should verify). To solve it, "partial integration" may be useful.
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
By differentiating with respect to x, we arrive at and we identify
we see that it is exact and hence partial integration can be used to solve it. Note that if
then by integrating partially with respect to , we have
for some function that does not depend on . If we differentiate
with respect to keeping constant then
from our initial observations. We therefore find
so that for an arbitrary constant C.
is the general solution to the ODE.