Question:
By eliminating the arbitrary constants from
\[
y=(a+b)\sin(x-c)-de^{x+e+f},
\]
the differential equation obtained is of order
By eliminating the arbitrary constants from
\[
y=(a+b)\sin(x-c)-de^{x+e+f},
\]
the differential equation obtained is of order
Show Hint
While finding the order of a differential equation formed by eliminating constants, count only the independent arbitrary constants. Constants appearing in combined form are treated as a single constant.
Updated On: Jun 22, 2026
- \(6\)
- \(4\)
- \(3\)
- \(5\)
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The Correct Option is C
Solution and Explanation
Step 1: Identify the arbitrary constants.
Given family of curves is \[ y=(a+b)\sin(x-c)-de^{x+e+f} \] Observe carefully that the constants appear in combined form: \[ (a+b)=A \] and \[ de^{e+f}=B \] Thus, the equation can be rewritten as \[ y=A\sin(x-c)-Be^x \] Now the arbitrary constants are \[ A,\quad B,\quad c \] Hence, there are \(3\) independent arbitrary constants.
Step 2: Use the theory of differential equations.
The order of the differential equation obtained after eliminating arbitrary constants is equal to the number of independent arbitrary constants present in the given family.
Since there are \(3\) independent arbitrary constants, the required differential equation will be of order \[ 3 \]
Step 3: Final conclusion.
Hence, the differential equation obtained is of order \[ \boxed{3} \]
Given family of curves is \[ y=(a+b)\sin(x-c)-de^{x+e+f} \] Observe carefully that the constants appear in combined form: \[ (a+b)=A \] and \[ de^{e+f}=B \] Thus, the equation can be rewritten as \[ y=A\sin(x-c)-Be^x \] Now the arbitrary constants are \[ A,\quad B,\quad c \] Hence, there are \(3\) independent arbitrary constants.
Step 2: Use the theory of differential equations.
The order of the differential equation obtained after eliminating arbitrary constants is equal to the number of independent arbitrary constants present in the given family.
Since there are \(3\) independent arbitrary constants, the required differential equation will be of order \[ 3 \]
Step 3: Final conclusion.
Hence, the differential equation obtained is of order \[ \boxed{3} \]
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