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Complete 8 of the problems on page 67 of the course text. Subproblems (a)-(i) of problem 9 can be considered separate problems.

For problem 1 of the homework, the form of the equation fits the criteria for using the Riccati equation.
Following through the notes starting on page 57, a function S(x) is introduced to mold the equation into the format of eq'n 2.15 to be solved with an integrating factor.
The problem I'm running into is that in order to find this integrating factor, the new "P(x)" term as seen in equation 2.15 consists of
(2P(x)S(x)+Q(x)) (see eq'n 2.31), and we do not know S(x) to be able to integrate it w.r.t x. How do we know what S(x) is?

irish1 wrote:For problem 1 of the homework, the form of the equation fits the criteria for using the Riccati equation.
Following through the notes starting on page 57, a function S(x) is introduced to mold the equation into the format of eq'n 2.15 to be solved with an integrating factor.
The problem I'm running into is that in order to find this integrating factor, the new "P(x)" term as seen in equation 2.15 consists of
(2P(x)S(x)+Q(x)) (see eq'n 2.31), and we do not know S(x) to be able to integrate it w.r.t x. How do we know what S(x) is?

The difficulty with the approach is that you have to know an S(x) a priori. In other words, given any solution, the book presents a method to find a general solution. Off the top of my head, I don't see a way to find such an S(x) except to guess.

The other option is to try to find an integrating factor.

If anyone has completed this problem and wants to at least indicate the approach they used, they may do so.

goodwine wrote:

irish1 wrote:For problem 1 of the homework, the form of the equation fits the criteria for using the Riccati equation.
Following through the notes starting on page 57, a function S(x) is introduced to mold the equation into the format of eq'n 2.15 to be solved with an integrating factor.
The problem I'm running into is that in order to find this integrating factor, the new "P(x)" term as seen in equation 2.15 consists of
(2P(x)S(x)+Q(x)) (see eq'n 2.31), and we do not know S(x) to be able to integrate it w.r.t x. How do we know what S(x) is?

The difficulty with the approach is that you have to know an S(x) a priori. In other words, given any solution, the book presents a method to find a general solution. Off the top of my head, I don't see a way to find such an S(x) except to guess.

The other option is to try to find an integrating factor.

If anyone has completed this problem and wants to at least indicate the approach they used, they may do so.

Well, I just figured it out. S(x) is just about as simple as it can get and is "guessable."