Differential math 2 Solve te wlli differ whle7w Bern yL x o

2, Solve te wlli differ wh-le7w Bern y\'--~-L x> o, Ajy

Solution

Solution:

y\' -y/x = -(*1.y²)/x : x>0

A Bernoulli differential equation

is of the form dy/dx + P(x)y = Q(x)yⁿ

here, P(x) = -1/x and Q(x)= -1/x and n=2

Let u = y^(1-n)

Then transforms the Bernoulli equation into the linear equation

du/dx + (1-n)P(x)u = (1-n)Q(x)

hence we get

du/dx + u/x = 1/x

Solve this equation by first order linear differential equation.

take u = a.b

so du/dx = a.db/dx + b.da/dx

put a.db/dx + b.da/dx = -(a.b)/x + 1/x

a(db/dx + b/x) + bda/dx -1/x=0

put a term =0 solution is db/dx + b/x=0 hence

-db/b= dx/x then integrate lnb = lnx + k so b= kx

hence bda/dx = 1/x put b=kx and integrate.

hence Solved.