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: Understanding the Question:
The question asks for the order of a differential equation whose general primitive solution is given as $y = a \cos x + b \sin x + c e^{-x}$.

Step 2: Key Formula or Approach:
The order of a differential equation is fundamentally equal to the total number of independent arbitrary constants (parameters) present in its general solution. If a solution contains $n$ essential independent arbitrary constants, the resulting differential equation obtained by eliminating them will have an order equal to $n$.

Step 3: Detailed Explanation:
Let's analyze the given general equation: $$y = a \cos x + b \sin x + c e^{-x}$$ In this equation, the coefficients $a$, $b$, and $c$ are distinct arbitrary constants that cannot be simplified or combined algebraically into fewer parameters.
Counting them directly:

• First constant: $a$

• Second constant: $b$

• Third constant: $c$
Since there are exactly 3 essential independent arbitrary constants in this relation, we must differentiate the function exactly 3 times to eliminate them all. Consequently, the corresponding differential equation is of the third order.

Step 4: Final Answer:
The order of the differential equation is 3, which corresponds to option (A).