Question

(xy^2+x)dx=x^2y+ydySolve for differential equation​

Answer 1

Step-by-step explanation:

We have,

$(x {y}^{2} + x)dx = ( {x}^{2} y + y)dy$

$\implies \: x( {y}^{2} + 1)dx = y( {x}^{2} + 1)dy$

$\implies \frac{2x}{ {x}^{2} + 1} dx = \frac{2y}{ {y}^{2} + 1 } dy$

Integrating both sides,

$\implies \int\frac{2x}{ {x}^{2} + 1} dx = \int\frac{2y}{ {y}^{2} + 1 } dy$

$\implies ln( {x}^{2} + 1) = ln( {y}^{2} + 1) + c$

$\implies {x}^{2} + 1 = {e}^{c}( {y}^{2} + 1)$

$\implies \frac{ {x}^{2} + 1 }{ {y}^{2} + 1 } = k$