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 derivative of sinh(x) is cosh(x), and the hyperbolic identity cosh^2(x) − sinh^2(x) = 1 immediately fits the differential equation.

1. Relevant knowledge points and concepts
- Derivative rules: d/dx[sinh x] = cosh x, d/dx[ln x] = 1/x, d/dx[e^x] = e^x, d/dx[cos x] = −sin x.
- Hyperbolic identity: cosh^2 x − sinh^2 x = 1 for all real x.
- The differential equation requires (dy/dx)^2 − y^2 to be identically equal to 1.

2. Complete reasoning process step by step
Test each candidate function by computing y, dy/dx, then (dy/dx)^2 − y^2 and check whether it equals 1 for all x in the domain.

3. Calculations (intermediate steps shown)
- For y = sinh x:
  dy/dx = cosh x
  (dy/dx)^2 − y^2 = cosh^2 x − sinh^2 x = 1 (by hyperbolic identity).
- For y = ln x:
  dy/dx = 1/x
  (dy/dx)^2 − y^2 = (1/x^2) − (ln x)^2, which is not identically 1 (depends on x and domain x>0).
- For y = e^x:
  dy/dx = e^x
  (dy/dx)^2 − y^2 = e^{2x} − e^{2x} = 0, not 1.
- For y = cos x:
  dy/dx = −sin x
  (dy/dx)^2 − y^2 = sin^2 x − cos^2 x = −cos(2x), not equal to 1 for all x.

4. Why the correct answer is correct
The hyperbolic identity gives an exact, x-independent value 1 for cosh^2 x − sinh^2 x, so y = sinh x satisfies the differential equation everywhere on its domain.

5. Why the other options are incorrect
- ln(x): expression (1/x^2) − (ln x)^2 varies with x and is not constant 1.
- e^x: yields 0, since derivative and function are equal, so squares cancel.
- cos x: yields −cos(2x), a nonconstant function, not equal to 1.

6. Conclusion
Thus the correct solution among the choices is a. y = sinh(x), which matches the provided answer.

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

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