Question

The order of the differential equation whose solution is $y = a \cos x + b \sin x + c e^{-x}$ is

• A. 3
• B. 4
• C. 2
• D. 1

Hint:

To quickly find the order of a general solution, always count the number of separate constants. Watch out for trick equations where constants can be combined (e.g., $y = a e^{x+b}$ contains two labels but can be rewritten as $y = (ae^b)e^x = C e^x$, which has only 1 independent constant)!

Answer 1

The Correct Option is A

Step 1: Recall the governing principle.
The order of a differential equation obtained by eliminating arbitrary constants equals the number of independent arbitrary constants in the general solution.
Step 2: Inspect the solution.
$y=a\cos x+b\sin x+c\,e^{-x}$ contains the constants $a,\,b,\,c$.
Step 3: Confirm independence.
The functions $\cos x$, $\sin x$, $e^{-x}$ are linearly independent, so none of $a,b,c$ can be absorbed into the others; all three are essential.
Step 4: Count them.
There are exactly $3$ independent arbitrary constants.
Step 5: Link count to order.
Eliminating $3$ constants requires differentiating three times, producing a third-order equation.
Step 6: State the answer.
Hence the order is $3$. \[ \boxed{\text{Order}=3} \]