I'm confused about the use of "won't" in a "when" sentence, I've seen many sentences like:
What to do When Your Dog Won't Eat
is that right? or should it be:
What to do When Your Dog doesn't Eat
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Let us look at this sentence:
When will you go to the museum?
When do you go to the museum?
Firstly, I hope that the first sentence does not sound odd to you. If it does, think of it this way. Can the second sentence replace the first? I say no. The first sentence is asking the person being questioned when he/she would go to the museum. Here, going to the museum is more like a site seeing, not a habitual one. While the second question is posed to (say), someone who works at the museum on regular basis.
With this ground, I will move forward to your question. In your examples, you are only adding the word 'not'. So, grammatically, both are correct.
But for your specific case, 'What to do when your dog won't eat?' is more suitable. The dog is not yours and you are only stating the possibility of the dog not eating.
Also, 'What to do when my dog doesn't eat?' is the better option you would have. You are saying that your dog often does not eat, i.e., you are here implying it to be your dog's habit not to eat.
I hope that clears up your confusion.
We use the verb will (negated, "won't" or past tense "wouldn't") in situations where the thing persists in "refusing" to do what it really should do, what it is expected to do:
My car won't start.
The bolt wouldn't budge so I had to take a blowtorch to it.
My dog won't eat.
Most speakers would understand you to mean that you have made several attempts to start the car and several attempts to get the bolt to budge and have placed several bowls of food in front of the dog. But the car and the bolt and the dog are not cooperating with you, as it were.
We use doesn't with situations where something is simply defective
This phone doesn't work. I would like a replacement or a refund.
There's no implication in that simple "does not" that you have made attempts to get it to work. It is a simpler statement without the implications of "won't".
You can also use "doesn't" when the thing or creature does not have a particular capability or does not do a certain thing "by design":
This dog doesn't bark. It is a Basenji, the so-called "barkless dog".
This car was made in 2005. It doesn't park itself and it doesn't warn you when you stray out of your lane.