solve the following 1st order ODE

Question

Use an appropriate method to solve the following 1st order ODE:

$LaTeX: \left( x + \tan^{-1} y\right) d x \, + \, \frac{x + y}{1 + y^2} \cdot d y = 0$

Express the final answer in an explicit form if possible, if not, leave the final answer in an implicit form. Please show ALL work.

Yefim S. · Accepted Answer

This is exact equation, because ∂/∂y(x + tan^-1y) = ∂/∂x((x + y)/(y²+ 1) = 1/(1 + y²);

F(x,y) = ∫(x + tan^-1y)dx = x²/2 + xtan^-1y + φ(y);

∂F/∂y = x/(1 + y²) + φ'(y) = (x + y)/(1 + y²); φ'(y) = y/(1 + y²); φ(y) = ∫y/(1+ y²)dy = 1/2ln(1 +, y²) + C;

F(x,y) = x²/2 + xtan^-1y + 1/2ln(1 + y²) + C.

1 Expert Answer