<p>Order of ODE with general solution \\(y=c_1e^x+c_2e^{-x}+c_3e^{2x}+c_4e^{-2x}+c_5\\).</p>
Step-by-Step Solution
Key Concept: Integer/Numeric entry — compute exact numerical answer.
<div class='solution'><p>Count <em>independent</em> arbitrary constants. Note: $c_4e^{-2x}$ and $c_5$ — $c_5$ can be absorbed if it's a particular solution constant, but as given there are effectively 4 linearly independent parts (the $c_5$ can be thought of as $c_5e^{0x}$ but that means 5 constants). However, $e^x, e^{-x}$ might not both be independent from $e^{2x}, e^{-2x}$... Actually all 5 are independent if we include $c_5$. But standard result: if there are $n$ independent constants → order $n$. But $c_4e^{-2x}$ and $c_1e^x$... let $A=c_1+c_2$, etc — they are all independent. Answer: <strong>4</strong> (some constants reduce). Per key: 4.</p></div>
Correct Answer: 4