Gotenks99
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Problem 1: Consider the (nonlinear) differential equation
[tex]y' = \frac{ - y(y + 1)}{x} [/tex]

Without solving the equation, check which of the following are solutions:

[tex]y(x) = {x}^{2} - 1[/tex]
[tex]y(x) = \frac{2}{x - 2} [/tex]

That is: do not use the method of integrating factors to solve. Just check by plugging in.​

Respuesta :

LammettHash

If y = x² - 1, then y + 1 = x² and y' = 2x. Then in the DE, we would have

2x = - (x² - 1) x² / x   ⇒   2 = - x² + 1

but this only true for some values of x. So the first choice is not correct.

If y = 2/(x - 2), then y + 1 = x/(x - 2) and y' = -2/(x - 2)². In the DE,

-2/(x - 2)² = - (2/(x - 2)) (x/(x - 2)) / x   ⇒   -2/(x - 2)² = -2/(x - 2)²

so the second choice is the correct answer.

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