Find the general solution for 2x dy\dx + y 6x^3 differential equations.

Question

Find the general solution for differential equations.

\(2x\frac{dy}{dx} +y =\,6x^3\)

2x dy\dx + y 6x³

Answer 1

Given Differential Equation :

\(2x\frac{dy}{dx} +y =\,6x^3\)

Formula :

i) \(\int \frac {1}{x}\) dx = log x

ii) \(\int x^n dx\) \(\frac {x^{n+1}}{n+1} +c\)

iii) a log b = log b^a

vi) a ^log_a^b = b

v) General solution :

For the differential equation in the form of

\(\frac {dy}{dx} +py =Q\)

The general solution is given by,

y. (I. F.) = \(\int\) Q (I. F.) dx + c

Where integrating factor,

I. F. = \(e^{\int p\,dx}\)

Given differential equation is

\(2x \frac {dy}{dx} + y 6^3\)

Dividing the above equation by 2x,

\(\frac {dy}{dx}+\frac{1} {2x}. y \,3x^2 \) ………eq(1)

Equation (1) is of the form

\(\frac {dy}{dx} +py =Q\)

Where, \(P = \frac {1}{2x} \) and = 3x²

Therefore, integrating factor is