Chapter One first order differential equation - Homogenous
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2.2. Homogenous method
A function 𝑓( 𝑥,𝑦) is said to be homogeneous of degree n if the equation
𝑓( 𝑧𝑥,𝑧𝑦)= 𝑧𝑛 𝑓( 𝑥, 𝑦)
Holds for all 𝑥, 𝑦 , and 𝑧 (for which both sides are defined).
Example 1: 𝑓( 𝑥, 𝑦) = 𝑥 2+ 𝑦2 is homogeneous of degree 2
Example 2: 𝑓( 𝑥, 𝑦) = √𝑥 8− 3𝑥2 𝑦6 is homogeneous of degree 4
Example 3: 𝑓( 𝑥,𝑦) = 𝑥 3sin𝑦
𝑥 + 𝑥 2𝑦 ln𝑥 is homogeneous of degree 3
Example 4: 𝑓( 𝑥,𝑦) = 𝑦
𝑥 𝑒 𝑥+ tanh 𝑦
𝑥 is homogeneous of degree 0
The method for solving homogeneous equations follows from this fact:
The substitution y = x u dy = xdu + udx
Or x = y u dx = ydu + udy
Transforms a homogeneous equation into a separable one
Example 1: Solve
( 𝐱𝟐 – 𝐲𝟐 ) 𝐝𝐱 + 𝐱𝐲 𝐝𝐲 = 𝟎
Solution
This equation is homogeneous. Thus to solve it, make the substitutions
y = x u 𝑎𝑛𝑑 dy = x du + u dx:
( x2 – y2 ) dx + xy dy = 0
[ x2 – (xu)2] dx + x(xu) (x du + u dx) = 0
[ x2 – x2 u2] dx + (x3u du + x2 u2 dx) = 0
x2 dx – x2 u2 dx + x3u du + x2 u2 dx = 0
x2 dx + x3u du = 0 𝑑𝑖𝑣𝑖𝑑𝑖𝑛𝑔 𝑏𝑦 x2 𝑇ℎ𝑒𝑛 dx + xu du = 0
This final equation is now separable. Proceeding with the solution,
dx
x = − u du
∫dx
x = − ∫ u du → 1
2 u2= −lnx + lnC or u2=2lnC
x
Put u = y
x Then [y
x]2=2lnC
x or y2= 2 x2 lnC
x
