Question

Find the general solution for differential equations.

\(xdy - (y+2x^2)\, dx = 0\)

xdy - (y + 2x²) dx = 0

Answer 1

Given Differential Equation :

\(xdy - (y+2x^2)\, dx = 0\)

Formula :

i) \(\int \frac{1}{x}\) dx = log x

ii) alog b = log b^a 

iii) a^{log_a b} = b 

iv) \(\int\) 1 dx = x

v) 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, \(P = \frac{-1}{x}\, and\, Q = 2x\)

Therefore, integrating factor is

General solution is

Multiplying above equation by x,

∴ y = 2x² + cx

Therefore general equation is

y = 2x² + cx