Given Differential Equation :
\(xdy + ( y- x^3)\,dx = 0\)
Formula :a
i) \(\int \frac{1}{x}\) dx = log x
ii) aloga b = b
iii) \(\int x ^n\,dx = \frac{x^{n+1}}{n+1}\)
iv) General solution :
For the differential equation in the form of
\(\frac{dy}{dx}\, + \, Py = Q
\)
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
Equation (1) is of the form
\(\frac{dy}{dx} \, + Py\, = Q\)
Where, \(P= \frac{1}{x}\, and \, Q = x^2\)
Therefore, integrating factor is
General solution is
Dividing above equation by x,
∴ \(y = \frac{x^3}{4} + \frac{c}{x}\)
Therefore general equation is
\(y = \frac{x^3}{4} + \frac{c}{x}\)
