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Consider the following statements: Statement 1: The general solution of the differential equation dy/dx + y = 1 is y = e^x + c. Statement 2: For a differential equation, the number of arbitrary constants in its general solution equals the order of the equation.
- Statement 1 is false, while Statement 2 is true
- Both Statement 1 and Statement 2 are true, and Statement 2 correctly explains Statement 1
- Both Statement 1 and Statement 2 are true, but Statement 2 does not correctly explain Statement 1
- Statement 1 is true, while Statement 2 is false
Correct answer: Both Statement 1 and Statement 2 are true, but Statement 2 does not correctly explain Statement 1
Solution
Statement 1 is true because the general solution of the given first-order linear differential equation is indeed y = e^x + c, where c is an arbitrary constant. Statement 2 is also true as it correctly states that the general solution of a differential equation contains as many arbitrary constants as the order of the equation, but it does not provide a direct explanation for the specific solution in Statement 1.
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