Show that tan⁻¹ x + tan⁻¹ y = C is the general solution of the differential equation (1 + x²) dy +…

Show that $\tan^{-1} x + \tan^{-1} y = C$ is the general solution of the differential equation $(1 + x^2)\,dy + (1 + y^2)\,dx = 0$

Given: $(1 + x^2)\,dy + (1 + y^2)\,dx = 0$

To show: $\tan^{-1} x + \tan^{-1} y = C$

Answer

Proof that $\tan^{-1}x+\tan^{-1}y=C$ solves the equation

1. $(1+x^2)dy+(1+y^2)dx=0$.
2. Separate the variables: $\frac{dy}{1+y^2}+\frac{dx}{1+x^2}=0$ (dividing by $(1+x^2)(1+y^2)$).
3. Integrate both sides: $\int\frac{dy}{1+y^2}+\int\frac{dx}{1+x^2}=\int0$.
4. $\tan^{-1}y+\tan^{-1}x=C$, where $C$ is the constant of integration.
5. Hence $\tan^{-1}x+\tan^{-1}y=C$ is the general solution.