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Consider the following statements: Statement 1: A differential equation of the type y f(xy) dx + y dy = 0 can be transformed into a homogeneous equation by using the substitution y = vx. Statement 2: Every first-order, first-degree differential equation becomes homogeneous when we substitute y = vx. Choose the correct option.
- Statement 1 is true, Statement 2 is a valid explanation of Statement 1
- Statement 1 is true, Statement 2 is true; however, Statement 2 does not explain Statement 1
- Statement 1 is false, Statement 2 is true
- Statement 1 is true, Statement 2 is false
Correct answer: Statement 1 is true, Statement 2 is true; however, Statement 2 does not explain Statement 1
Solution
Statement 1 is accurate because the substitution y = vx can indeed transform the given differential equation into a homogeneous form. However, Statement 2 is broader and applies to all first-order, first-degree equations, making it true but not specifically explanatory for Statement 1.
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