Question

Solve the equation.

(tan^-1y – x)dy = (1 + y²)dx

Answer 1

The given differential equation is (tan^-1y – x)dy = (1 + y²)dx

Now (1) is a linear differential equation of the form dy/dx + P₁x = Q₁, where P₁ = 1/1+y² and Q₁ = \(\frac{tan^1y}{1+y^2}\)

Thus, the solution of the given differential equation is

Substituting tan^-1 y = t so that \((\frac{1}{1+y^2})\) dy = dt, we get

Substituting the value of I in equation (2), we get

which is the general solution of the given differential equation.