Question

The elimination of arbitrary constants $c_{1}, c_{2}, c_{3}, c_{4}$ from $y=(c_{1}+c_{2})\sin(2x+c_{3}) + c_{4}e^{5x}$ gives a differential equation of order ________.

• A. 2
• B. 4
• C. 3
• D. 1
• E. 5

Hint:

Combine dependent constants (like $c_1+c_2$) 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 from a given relation is equal to the number of \textit{essential} or \textit{independent} arbitrary constants in that relation. We need to identify how many independent constants are actually present in the given equation.
Step 2: Key Formula or Approach
1. Examine the given equation for arbitrary constants.
2. Check if any constants can be combined or are dependent on each other.
3. The number of independent constants determines the order of the resulting differential equation.
Step 3: Detailed Explanation
1. Analyze the given equation.
The equation is \(y = (c_1 + c_2)\sin(2x + c_3) + c_4e^{5x}\).
The arbitrary constants are listed as \(c_1, c_2, c_3, c_4\).
2. Identify the independent constants.
- In the first term, we have the expression \((c_1 + c_2)\). Since \(c_1\) and \(c_2\) are arbitrary constants, their sum \((c_1 + c_2)\) is also just a single arbitrary constant. Let's call it \(A = c_1 + c_2\).
- The constant \(c_3\) is inside the sine function and acts as a phase shift. It is independent of A. Let's call it \(B = c_3\).
- The constant \(c_4\) is the coefficient of the exponential term. It is independent of the others. Let's call it \(C = c_4\).
The equation can be rewritten as:
\[ y = A \sin(2x + B) + C e^{5x} \] We can expand the sine term using the sum formula: \(\sin(U+V) = \sin U \cos V + \cos U \sin V\).
\[ y = A (\sin(2x)\cos B + \cos(2x)\sin B) + C e^{5x} \] \[ y = (A \cos B) \sin(2x) + (A \sin B) \cos(2x) + C e^{5x} \] Let \(A' = A \cos B\) and \(A'' = A \sin B\). Now the equation is:
\[ y = A' \sin(2x) + A'' \cos(2x) + C e^{5x} \] Here, \(A'\), \(A''\), and \(C\) are our three independent arbitrary constants.
- \(A' = (c_1+c_2)\cos(c_3)\)
- \(A'' = (c_1+c_2)\sin(c_3)\)
- \(C = c_4\)
The number of independent arbitrary constants is 3.
3. Determine the order of the differential equation.
The order of the differential equation formed by eliminating the arbitrary constants is equal to the number of independent arbitrary constants.
Since there are 3 independent constants, the order of the differential equation will be 3.
Step 4: Final Answer
The differential equation will be of order 3.