​ Find the general solution for differential equations. (x + 3y^3) dy/dx = y. (y > 0) ​

Question

Find the general solution for differential equations.

 (x + 3y³) \(\frac{dy}{dx}\) = y. (y > 0)

Answer 1

Given Differential Equation :

(x + 3y³) \(\frac{dy}{dx}\) = y

Formula :

i) \(\int \frac {1}{x}\) dx = log x

ii) a log b = log b^a

iii) a^{log_a b} = b 

iv) \(\int\) xⁿ dx = \(\frac {x^{n+1}}{n+1}\)

v) General solution :

For the differential equation in the form of

\(\frac{dx}{dy} \, + Px\, =Q\)

General solution is given by,

x. (I.F.) = \(\int\) Q. (I.F.) dy + c

Where, integrating factor,

I.F. = \(e^{\int P\, dy}\)

Given differential equation is

Equation (1) is of the form

\(\frac{dx}{dy} \, + Px\, =Q\)

where, P = \(\frac {-1}{y}\) and Q = 3y²

Therefore, integrating factor is

General solution is

Multiplying above equation by y,

Therefore, general solution is