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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 false
Solution
Statement 2 is false: not every first-order first-degree equation becomes homogeneous under y=vx (that substitution works only for already-homogeneous equations). Statement 1, that the y f(xy) form can be reduced to a homogeneous equation, is the true claim. Hence Statement 1 true, Statement 2 false.
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