Question

b) Solve the Bernoulli equation by reducing it to
linear differential equation.
(xy5 + y)dx – dy = 0​

Answer 1

Given:

(xy⁵ + y)dx – dy = 0​

To find:

Reducing the equation to  linear differential equation.

Solution:

$(xy^{5} + y )dx = dy =0$

$(xy^{5}+y )dx = dy.$

$xy^{5} dx + ydx = dy$.

$\int\lx x(y^{5} )dx +\int\ y^{5}(x)dx + \int\ ydx = \int\ dy.$

$x^{2} y^{5} } + \frac{x^{2} y^{5} }{2} + xy = y$.

$\frac{2x^{2} y^{5}+ x^{2} y^{5}}{2} + xy = y$.

$\frac{3x^{2} y^{5} }{2} + xy = y$.

​$3x^{2} y^{5} + 2xy = 2y$.

$3x^{2} y^{4} + 2x = 2.$

Hence, the above equation is reduced to $3x^{2} y^{4} + 2x = 2.$