Question

John Bernoulli found that a differential equation to the brain his to chrone problem is y[1+(dy/dx)²]=c Find equation of x and y in terms of another variable

Answer 1

The equation representing the solution to John Bernoulli's differential equation is
`\( x = A \cos(t) \) and \( y = c - A \sin(t) \), where \( A \)` is an arbitrary constant and
`\( t \)` is a parameter.

Step-by-step explanation:

John Bernoulli's differential equation,
`\( y[1 + (dy/dx)^2] = c \)`, is a first-order nonlinear ordinary differential equation. To find the solution, we can use the substitution
`\( dy/dx = \tan(t) \), where \( t \)` is a parameter. This transforms the differential equation into a separable one.

Integrating with respect to
`\( t \) yields \( y = c - A \sin(t) \), where \( A \)` is an arbitrary constant. To find
`\( x \), we use the fact that \( \tan(t) = dy/dx \), leading to \( x = A \cos(t) \).`

The solution represents parametric equations for the curve. As
`\( t \)`varies, the corresponding values of
`\( x \) and \( y \)` trace the curve determined by the given differential equation. The arbitrary constant
`\( A \)` allows for different parameterizations of the curve. This solution provides insight into the relationship between
`\( x \) and \( y \)`that satisfies the given differential equation.