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# Find all solutions of the differential equation y′ sin x + y cos x = 1

Find all solutions of the following differential equation on the interval : Prove that exactly one of these solutions has a finite limit as , and that exactly one of these solutions has a finite limit as .

Proof. First, to find the solutions of the differential equation we divide by (noting for any ), We apply Theorem 8.3 (page 310 of Apostol) with Then, we choose and find solutions in terms of . This gives us Therefore, where . These are all of the solutions on the given interval.

Next, to find the solutions with finite limits as and . If is finite, we must have (otherwise the limit will go to infinity). Therefore, .
If is finite, we must have . Therefore, 1. anonymouse says:

How did you figure out to use a = pi / 2 ?

• Anonymous says:

A very late response, but the reason is having cos(a)=0 makes calculations easier.

• Anonymous says:

I’m sorry – I meant having ln(sin(a))=0 (which is the case here since sin(pi/2)=1 and ln(1)=0) makes calculations easier :) If I’m reasoning correctly, of course