Question

Find the general/particular solution of the following Differential Equations

(Non-Exact D.E)

(y - xy³)dx + xdy=0

Answer 1

Rewrite in the form of a first order Bernoulli ODE

$$y'+\dfrac{1}{x}y=y^3$$

Substitute $v=y^{1-3}$

$$v'=-\dfrac{2}{y^3}y'$$

$$-\dfrac{1}{2}v'+\dfrac{1}{x}v=1$$

Integrating factor $\mu(x)=\dfrac{1}{x^2}$

$$\dfrac{1}{x^2}v'-\dfrac{2}{x^3}v=-\dfrac{2}{x^2}$$

$$(\dfrac{v}{x^2})'=-\dfrac{2}{x^2}$$

$$\int d(\dfrac{v}{x^2})=-\int\dfrac{2}{x^2}dx$$

$$\dfrac{v}{x^2}=\dfrac{2}{x}+C$$

$$v=2x+Cx^2$$

Substitute back $v=y^{-2}$

$$y^{-2}=2x+Cx^2$$

$$y^2=\dfrac{1}{2x+Cx^2}$$