Problem 1: Consider the (nonlinear) differential equation y' = ( - y(y + 1))/(x) Without solving the equation, check which of the following are solutions: y(x) = {x}^(2) - …

Question

Problem 1: Consider the (nonlinear) differential equation


`y' = ( - y(y + 1))/(x)`

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


`y(x) = {x}^(2) - 1`

`y(x) = (2)/(x - 2)`

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

Answer 1

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.