Question

Consider the differential equation


`y' = {x}^(2) (1 + {y}^(2) ).`
Which of the following are solutions:


`y(x) = {x}^(2) + 3`


`y(x) = tan( \frac{x {}^(3) }{3} + 2).`


`y(x) = \sqrt{ {x}^(2) + 1 } .`

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

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Answer 1

Answer: For this problem I used the separation method instead of plugging in each one. I thought this would be easiereasier.to

Explanation:

Answer 2

The first solution is quadratic, so its derivative y' on the left side is linear. But the right side would be a polynomial of degree greater than 1, so this is not the correct choice.

The third solution has a similar issue. The derivative of √(x² + 1) will be another expression involving √(x² + 1) on the left side, yet on the right we have y² = x² + 1, so that the entire right side is a polynomial. But polynomials are free of rational powers, so this solution can't work.

This leaves us with the second choice. Recall that

1 + tan²(x) = sec²(x)

and the derivative of tangent,

(tan(x))' = sec²(x)

Also notice that the ODE contains 1 + y². Now, if y = tan(x³/3 + 2), then

y' = sec²(x³/3 + 2) • x²

and substituting y and y' into the ODE gives

sec²(x³/3 + 2) • x² = x² (1 + tan²(x³/3 + 2))

x² sec²(x³/3 + 2) = x² sec²(x³/3 + 2)

which is an identity.

So the solution is y = tan(x³/3 + 2).