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:
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:
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Always simplify constants (like $c_1+c_2$ or $e^{c}$) into a single constant before counting.
Updated On: Apr 28, 2026
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The Correct Option is C
Solution and Explanation
Step 1: Concept
The order of a differential equation is equal to the number of independent arbitrary constants in the general solution.
Step 2: Analysis
The expression is $y = (c_1 + c_2)\sin(x + c_3) - c_4e^x$. Combine $(c_1 + c_2)$ into a single constant $K$. So, $y = K\sin(x + c_3) - c_4e^x$.
Step 3: Conclusion
The independent constants are $K, c_3,$ and $c_4$. Since there are 3 independent constants, the order of the differential equation is 3. Final Answer: (C)
The order of a differential equation is equal to the number of independent arbitrary constants in the general solution.
Step 2: Analysis
The expression is $y = (c_1 + c_2)\sin(x + c_3) - c_4e^x$. Combine $(c_1 + c_2)$ into a single constant $K$. So, $y = K\sin(x + c_3) - c_4e^x$.
Step 3: Conclusion
The independent constants are $K, c_3,$ and $c_4$. Since there are 3 independent constants, the order of the differential equation is 3. Final Answer: (C)
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