Question

Obtén la solución general de la siguiente ecuación diferencial separable: (x + xy²) dx + ex²y dy = 0​

Answer 1

Respuesta:

Explicación paso a paso:

(x + xy²) dx + e²y dy = 0

(x + xy²) dx = - ex²y dy

x(1 + y²) dx = - ex²y dy

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

∫$\frac{1}{ x } dx$ = - e∫$\frac{y}{1+y^{2} }dy$

|Ln x| = -e/2 Ln|1+y²| + C    como 1+y² >0 |1 + y²| = (1 + y²)

|Ln x| = -e/2 Ln (1+y²) + Ln ($e^{C}$)

|Ln x| = Ln $(1+y^{2} )^{-\frac{e}{2} }$ +  Ln ($e^{C}$)

|Ln x| = Ln [$(1+y^{2} )^{-\frac{e}{2} }.e^{C}$ ]

x = $(1+y^{2} )^{-\frac{e}{2} }.e^{C}$    ;    $e^{C}$ = K

$X =\frac{k}{ (1+y^{2})^{\frac{e}{2} }}$