Question

Please answer fast
Q4.

Answer 1

Answer:

either this way or wait i will have to type a solution

Step-by-step explanation:

The given homogeneous differential equation is

(x²y - 2xy²) dx - (x³ - 3x²y) dy = 0

or, dy/dx = (x²y - 2xy²)/(x³ - 3x²y)

or, dy/dx = {(x/y)² - 2 (x/y)}/{(x/y)³ - 3 (x/y)²}

[ dividing numerator and denominator by y³ ]

Let y = vx. Then dy/dx = v + x dv/dx

Continuing,

v + x dv/dx = (1/v² - 2/v)/(1/v³ - 3/v²)

or, v + x dv/dx = (v - 2v²)/(1 - 3v)

or, x dv/dx = (v - 2v² - v + 3v²)/(1 - 3v)

or, x dv/dx = v²/(1 - 3v)

or, (1 - 3v)/v² * dv = dx/x

or, dv/v² - 3 dv/v = dx/x

On integration, we get

∫ dv/v² - 3 ∫ dv/v = ∫ dx/x

or, - 1/v - 3 logv = logx + logc [ logc = int. const. ]

or, - 1/v - log(v³) = log(cx)

or, 1/v = - {log(v³) + log(cx)}

or, 1/v = - {log(cxv³)}

or, x/y = - {log(c * x * y³/x³)} [ ∵ v = y/x ]

or, x/y = - log(cy³/x²)

or, x/y = log(x²/cy³)

This is the required primitive. (Ans.)