Question

The order of the differential equation whose general solution is given by $y = (C_1 + C_2) \sin(x + C_3) - C_4 e^{x+C_5}$ is \dots}

• A. 5
• B. 4
• C. 2
• D. 3

Hint:

Do not just count the number of $C$'s blindly! Expressions like $(C_1 \pm C_2)$, $C_1 C_2$, $C_1/C_2$, and phase shifts like $\sin(x+C)$ or scaled exponents like $e^{x+C}$ ALWAYS compress down to fewer essential constants.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We are given the general solution of a differential equation containing 5 arbitrary constants ($C_1$ to $C_5$). We need to determine the order of the corresponding differential equation.

Step 2: Key Formula or Approach:
The order of a differential equation is strictly equal to the number of independent (essential) arbitrary constants present in its most simplified general solution.
We must use algebraic and trigonometric identities to compress the given constants into the minimum possible number of distinct parameters.

Step 3: Detailed Explanation:
Given equation:
$$y = (C_1 + C_2) \sin(x + C_3) - C_4 e^{x+C_5}$$
1. Compress $C_1 + C_2$:
Let $A = C_1 + C_2$, where $A$ is a single new constant.
$$y = A \sin(x + C_3) - C_4 e^{x+C_5}$$
2. Expand the sine term using $\sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta$:
$$y = A (\sin x \cos C_3 + \cos x \sin C_3) - C_4 e^{x+C_5}$$
Distribute the $A$:
$$y = (A \cos C_3) \sin x + (A \sin C_3) \cos x - C_4 e^{x+C_5}$$
Let $P = A \cos C_3$ and $Q = A \sin C_3$, where $P$ and $Q$ are two new independent constants.
$$y = P \sin x + Q \cos x - C_4 e^{x+C_5}$$
3. Expand the exponential term using $e^{a+b} = e^a \cdot e^b$:
$$- C_4 e^{x+C_5} = - C_4 e^{C_5} \cdot e^x$$
Let $R = C_4 e^{C_5}$, where $R$ is a single new constant.
$$y = P \sin x + Q \cos x - R e^x$$
This final equation contains exactly 3 distinct, non-compressible arbitrary constants ($P$, $Q$, and $R$).
Therefore, the differential equation governing this family of curves must be of order 3.

Step 4: Final Answer:
The order is 3, which matches option (d).