I using php's str_replace function to replace some text. Example below.
str_replace("{x+{\$v}}", "{x-{\$v}}", $this->introtext);
str_replace('{x+{\$v}}', '{x-{\$v}}', $this->introtext);
In first case it replace text, but in second case it is not doing so. What is difference between two?
There is a difference between ' and ". Get rid of the \ inside the single-quoted string.
EDIT: To clarify, a double-quoted string does some expansion (see the link), whereas the single-quoted one does not. $ needs to be escaped with \ inside double-quoted strings, but not in single-quoted strings. In the latter case, it would literally result in \$.
The dollar sign and curly braces have special meanings when inside double-quoted strings. $ is used to put content from a variable directly into a string. {} is used to wrap PHP variables in case their meaning is ambiguous.
For example "$hello" would put the contents of the variable $hello into the string, whereas "{$hell}o" would put the contents of the variable $hell into the string, followed by the literal letter o.
Single quotes don't have this at all (and are generally preferable when quoting more complex strings, for simplicity). So '{$hell}o' would be a string containing the actual characters {$hell}o.
It's a little unclear what exact strings you are trying to replace. To replace the literal {x+$v} with {x-$v} then use:
str_replace('{x+$v}', '{x-$v}', $this->introtext);