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 Concept:
The order of a differential equation is equal to the number of independent arbitrary constants in its general solution.

Step 2: Formula Application:
Simplify the constants: - $(C_1 + C_2)$ is just one constant, let's call it $A$. - $e^{x+C_5} = e^x \cdot e^{C_5}$. So $C_4 e^{C_5}$ is another single constant, let's call it $B$.

Step 3: Explanation:
The equation becomes: $y = A \sin(x + C_3) - B e^x$. The remaining arbitrary constants are $A, C_3,$ and $B$. Total number of independent constants $= 3$.

Step 4: Final Answer:
The order of the differential equation is 3.