In problems 11-14 verify that the indicated function is an explicit solution of the given differential equation. Assume an appropriate interval I of definition for each solution.
y'' + y = tanx ; y = -cos(x)ln(sec(x) + tan(x))
Anyways, next problem...
In problems 15-18 verify that the indicated function y = g(x) is an explicit solution of the given first order differential equation. Proceed by giving g(x) simply as a function and give its domain. Then by considering g(x) as a solution of the differential equation, give at least one interval I of definition.
(y-x)y' = y-x+8 ; y = x+( 4(x+2)^(1/2) )
The Attempt at a Solution
My answer for question 13:
Here I verified that is is indeed an explicit solution by plugging in. I posted the derivatives because it might help answer my questions.
y=-cos(x)ln(sec(x) + tan(x))
y' = sin(x)ln(sec(x)+tan(x))-1
y'' = cos(x)ln(sec(x)+tan(x))+tan(x)
Moving on to the next part of the question, "Assume an appropriate interval I of definition for each solution."
This is how I understand this question: At the beginning of the question it says "verify that the indicated function is an explicit solution" and then "Assume an appropriate interval I of definition for each solution."
So the solution being mentioned here is just y(x). y(x) is the only function that I'm finding an interval of definition for. Even though the interval of definition is the same for y, y', and y'', I care about y(x) because that's my solution.
Can anyone tell me if my understanding is correct here?
Now my answer for question 15:
y = x+( 4(x+2)^(1/2) )
y' = ( (x+2)^(1/2) + 2 ) / (x+2)^(1/2)
I checked and it is indeed an explicit solution.
so for y = x+( 4(x+2)^(1/2) ) , you can plug in any number equal to or greater than -2 and you will still get a solution. So going with the logic of the question before this one, I chose my interval of definition to be [-2, infinity). But my book tells me that the largest interval of definition is actually (-2, infinity)
This leads me to believe my previous theory was wrong. The largest interval of definition would be [-2, infinity) if I am just taking y into account. But it looks like for y', it is (-2, infinity) and that's the answer.
and if I keep on taking derivatives of y after y', I always get a denominator of (x+2)^(n/2), meaning that its going to be (-2, infinity) for every derivative after y' as well.
If you guys can please answer the previous questions and this one final question:
so when finding the interval of definition, it is going to be the interval that satisfies every single dependent variable and all of its derivatives as well?
I know the question was really long guys, but its a simple diff eq question but my book and instructor just haven't been able to get the concept across to me... I appreciate the time guys and I truly do care about not wasting anyones time hopefully I can contribute to this community one day as well.