Supplementary Material
Linaer & Non-Linear DE
Dr. lory liza d. bulay-og, PECE
Associate Prof 1
Linearity
+ The important issue is how the unknown y appears in
the equation. A linear equation involves the
dependent variable (y) and its derivatives by
themselves. There must be no "unusual" nonlinear
functions of y or its derivatives.
+ A linear equation must have constant coefficients, or
coefficients which depend on the independent
variable (t). If y or its derivatives appear in the
coefficient the equation is non-linear.
+ A linear differential equation can be recognized
by its form. It is linear if the coefficients of y (the
dependent variable) and all order derivatives of
y, are functions of t, or constant terms, only are
all linear.
Linearity - Examples
The dependent variable is
y, the equation contains
Only the derivative of y with
respect to t and
Constant y, thus, it’s linear
dy
y0
dt
The dependent variable is
x, the equation contains 𝑥 2 ,
Thus non-linear
dx
 x2  0
dt
is linear
is non-linear
Linearity - Examples
The dependent variable is
y, the equation contains
Only the derivative of y with
respect to t and
Constant y, thus, it’s linear
The dependent variable is
y, the equation contains y multiplied
by derivative of y with respect to t,
Thus non-linear
dy 2
t  0
dt
y
dy 2
t  0
dt
is linear
is non-linear
More examples:
+
+
+
+
𝑑𝑦
𝑥 + 2𝑦 = 2𝑥
𝑑𝑥
𝑑𝑦
+ 𝑦𝑐𝑜𝑡𝑥 = 2𝑥 2
𝑑𝑥
𝑑𝑦
+ 𝑥2𝑦 = 𝑥
𝑑𝑥
1 𝑑2 𝑦
3 = 3𝑥
−
𝑦
𝑥 𝑑𝑥 2
𝑑𝑦
(both 𝑑𝑥 𝑎𝑛𝑑 𝑦 𝑎𝑟𝑒 linear, thus the equation is linear)
𝑑𝑦
(both 𝑑𝑥 𝑎𝑛𝑑 𝑦 𝑎𝑟𝑒 linear, thus the equation is linear)
𝑑𝑦
(both 𝑑𝑥 𝑎𝑛𝑑 𝑦 𝑎𝑟𝑒 linear, thus the equation is linear)
(the term 𝑦 3 is non linear, thus the equation is not linear)
+
𝑑3 𝑦
𝑑𝑥 3
𝑑2 𝑦
−2 2
𝑑𝑥
+
𝑑𝑦
𝑑𝑥
= 2𝑠𝑖𝑛𝑥
𝑑 3 𝑦 𝑑 2 𝑦 𝑑𝑦
(the term
,
&
𝑎𝑟𝑒 𝑎𝑙𝑙 linear, thus the equation is linear)
𝑑𝑥 3 𝑑𝑥 2 𝑑𝑥
+
𝑑𝑦
𝑑𝑥
− 𝑠𝑖𝑛𝑦 = −𝑥
( the term sin y is non-linear, thus the equation is not linear)
Linearity – Summary
Linear
2y
dy
dt
(2  3 sin t) y
dy
t
dt
Non-linear
y2
or
y
sin( y )
dy
dt
(2  3 y 2 ) y
 dy 
 
 dt 
2
Thanks!
Any questions?
You can find me at:
+ lory.bulay-og@ustp.edu.ph
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