Which of the following is a solution to the differential equation$$\left(\dfrac{dy}{dx}\right)^2-y^2=1$$

Question

BlackTom AI · Accepted Answer

The hyperbolic identity cosh^2(x) − sinh^2(x) = 1 together with the derivative d/dx[sinh x] = cosh x is the key fact.

1) Relevant concepts: derivative rules and hyperbolic trig identities. The differential equation requires (y')^2 − y^2 = 1 to hold for all x in the domain.

2) Check each option by computing y' and substituting into (y')^2 − y^2.

- d. y = sinh x:
  y' = cosh x, so (y')^2 − y^2 = cosh^2 x − sinh^2 x = 1 by the standard hyperbolic identity. Thus y = sinh x satisfies the equation identically.

- a. y = ln x:
  y' = 1/x, so (y')^2 − y^2 = 1/x^2 − (ln x)^2. This expression is not identically equal to 1 for all x (for example at x = 2 it equals 1/4 − (ln 2)^2 ≈ −0.231 ≠ 1). So ln x is not a solution.

- b. y = cos x:
  y' = −sin x, so (y')^2 − y^2 = sin^2 x − cos^2 x = −cos 2x. That equals 1 only at isolated x values (not identically), e.g. at x = 0 it gives −1 ≠ 1. So cos x is not a solution.

- c. y = e^x:
  y' = e^x, so (y')^2 − y^2 = e^{2x} − e^{2x} = 0, which is not 1. Thus e^x fails.

3) Intermediate steps shown above for each substitution; the only identity-producing substitution is sinh x.

4) Why the correct answer is correct: the hyperbolic identity guarantees cosh^2 x − sinh^2 x = 1 exactly, and since y' for sinh is cosh, the differential equation is satisfied pointwise for all x.

5) Other options fail because their substituted expressions do not simplify to the constant 1 for all x (examples given).

Conclusion: y = sinh(x) (option d) is the correct solution, matching the provided answer.

Which of the following is a solution to the differential equation$$\left(\dfrac{dy}{dx}\right)^2-y^2=1$$Single choice

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