Solving the Differential Equation for y x tan^-1 p

In this article, we will walk through the process of finding the differential equation corresponding to the given relation:

y x tan^-1 p

where p is typically a function of x and y.

Step-by-Step Solution

To find the differential equation, we need to differentiate the given relation with respect to x. Let's denote p dy/dx.

Initial Equation: y x tan^-1 p Step 1: Differentiate both sides of the equation with respect to x using the chain rule. Step 2: The derivative of tan^-1 p with respect to x is given by:

dy/dx 1 (1/(1 p^2)) * (dp/dx)

Step 3: Substitute back p dy/dx into the equation.

dy/dx 1 (1/(1 (dy/dx)^2)) * (d/dx(dy/dx))

Step 4: Simplify the equation:

dy/dx - 1 (1/(1 (dy/dx)^2)) * (d^2y/dx^2)

Step 5: Rearrange the equation to the standard form:

(1 (dy/dx)^2) * (dy/dx - 1) d^2y/dx^2

This is the differential equation relating y, p, and their derivatives.

Alternative Approach

Let's consider an alternative approach using the chain rule and trigonometric identities:

Initial Equation: y x tan^-1(p) Step 1: Express the equation in a different form:

y - x/a tan^-1(p)

Step 2: Differentiate both sides with respect to x using the chain rule:

d/dx(tan^-1(p)) d/dx(y - x/a)

Step 3: The derivative of tan^-1(p) with respect to x is:

1/(1 p^2) * dp/dx

Step 4: The right-hand side is:

dy/dx - 1/a

Step 5: Equate the derivatives:

1/(1 p^2) * dp/dx dy/dx - 1/a

Step 6: Rearrange the equation:

(1 p^2) * (dy/dx - 1/a) dp/dx

This gives us another form of the differential equation.

Conclusion

In conclusion, we have derived the differential equation corresponding to the given relation y x tan^-1(p). The final form of the equation is:

(1 (dy/dx)^2) * (dy/dx - 1) d^2y/dx^2

This equation can be used to solve problems involving the function y in terms of x and its derivatives.

Keywords: differential equation, tan^-1, calculus