Question

The elimination of arbitrary constants $c_{1}, c_{2}, c_{3}, c_{4}$ from $y=(c_{1}+c_{2})\sin(x+c_{3})-c_{4}e^{x}$ gives a differential equation of order:

• A. 1
• B. 2
• C. 3
• D. 4
• E. 5

Hint:

Always simplify constants (like $c_1+c_2$ or $e^{c}$) into a single constant before counting.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The order of a differential equation formed by eliminating arbitrary constants is equal to the number of essential arbitrary constants in the original equation. We need to identify how many independent constants are present in the given relation.
Step 2: Key Formula or Approach:
Analyze the given equation and identify the constants. If some constants can be combined into a single new constant, they are not independent. Count the number of independent, or essential, constants.
Step 3: Detailed Explanation:
The given equation is:
\[ y = (c_1+c_2)\sin(x+c_3) - c_4 e^x \] Let's examine the constants: \( c_1, c_2, c_3, c_4 \). 1. The term \( (c_1+c_2) \) is a sum of two arbitrary constants. This sum can be represented by a single new arbitrary constant. Let \( A = c_1 + c_2 \).
2. The constant \( c_3 \) is inside the sine function and cannot be combined with others. It is an essential constant.
3. The constant \( c_4 \) is a coefficient of \( e^x \) and cannot be combined. It is an essential constant.
The equation can be rewritten with the essential constants as:
\[ y = A \sin(x+c_3) - c_4 e^x \] The essential arbitrary constants are A, \( c_3 \), and \( c_4 \).
There are 3 essential arbitrary constants.
To eliminate 3 essential constants, we need to differentiate the equation 3 times, which will result in a differential equation of order 3.
Step 4: Final Answer:
The order of the resulting differential equation is 3.