This lecture will improve your knowledge in differential equation lecture is by Lecturer AZEEM INAAM of SMI university.

1. DIFFERENTIAL EQUATION (MT-202) SYED AZEEM INAM
DIFFERENTIAL EQUATION (MT-202)
LECTURE #4
TYPE 2 HOMOGENOUS EQUATIONS:
A function of each degree n is called a homogenous function of degree n.
TEST: Any function f(x, y) is said to be a homogenous function of degree n in x and
y, if
𝑓(𝑡𝑥, 𝑡𝑦) = 𝑡 𝑛
𝑓(𝑥, 𝑦)
The differential equation
𝑑𝑦
𝑑𝑥
= 𝑓(𝑥, 𝑦) → (1)
is said to be homogenous function of degree 0, i.e. if f(x, y) can be written in the
form g(y/x) or h(x/y). Such equation can be solved by substituting y=vx or x=vy. If
differential equation is homogenous then it is equivalent to
𝑑𝑦
𝑑𝑥
= 𝑔 (
𝑦
𝑥
) → (2)
Substituting y=vx in (2) we have
𝑣 + 𝑥
𝑑𝑣
𝑑𝑥
= 𝑔(𝑣) ↔ 𝑥
𝑑𝑣
𝑑𝑥
= 𝑔(𝑣) − 𝑣
This is now a variable separable equation and can be solved by variables separable
method. The solution is deduced by replacing ‘v’ from (y/x).