You will see many sites with the following type of code:
Script elements:
No, they aren't.
For a long time, browser sets the default type=text/javascript
at script elements, ect. when you omitted the type
attribute. HTML5 makes this official, but every browser supported it long time ago.
In fact, which doctype you use isn't affecting the browsers behaviour, only the W3C specs. The nice and short HTML5 doctype is the standard doctype for years in normal browsers. HTML5 just makes those nice things official.
So you can use it and no browser will scream or fail, only the validator.