This document discusses linearity in differential equations. It defines a linear differential equation as one where the dependent variable (y) and its derivatives appear alone and are not within nonlinear functions. The coefficients must also be constants or depend only on the independent variable (t). Examples are provided of both linear and nonlinear equations to illustrate these concepts. Key factors that determine if an equation is linear or nonlinear are whether y or its derivatives are raised to powers or involved in trigonometric functions.

1. Linaer & Non-Linear DE
Dr. lory liza d. bulay-og, PECE
Associate Prof 1
Supplementary Material

2. 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.

3. + 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.

4. Linearity - Examples
0

 y
dt
dy
is linear
0
2

 x
dt
dx
is non-linear
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
x, the equation contains 𝑥2
,
Thus non-linear

5. Linearity - Examples
is linear
0
2

 t
dt
dy
0
2

 t
dt
dy
y is non-linear
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

6. More examples:
+ 𝑥
𝑑𝑦
𝑑𝑥
+ 2𝑦 = 2𝑥 (both
𝑑𝑦
𝑑𝑥
𝑎𝑛𝑑 𝑦 𝑎𝑟𝑒 linear, thus the equation is linear)
+
𝑑𝑦
𝑑𝑥
+ 𝑦𝑐𝑜𝑡𝑥 = 2𝑥2
(both
𝑑𝑦
𝑑𝑥
𝑎𝑛𝑑 𝑦 𝑎𝑟𝑒 linear, thus the equation is linear)
+
𝑑𝑦
𝑑𝑥
+ 𝑥2𝑦 = 𝑥 (both
𝑑𝑦
𝑑𝑥
𝑎𝑛𝑑 𝑦 𝑎𝑟𝑒 linear, thus the equation is linear)
+
1
𝑥
𝑑2𝑦
𝑑𝑥2 − 𝑦3
= 3𝑥 (the term 𝑦3
is non linear, thus the equation is not linear)

7. +
𝑑3𝑦
𝑑𝑥3 − 2
𝑑2𝑦
𝑑𝑥2 +
𝑑𝑦
𝑑𝑥
= 2𝑠𝑖𝑛𝑥
(the term
𝑑3
𝑦
𝑑𝑥3
,
𝑑2
𝑦
𝑑𝑥2
&
𝑑𝑦
𝑑𝑥
𝑎𝑟𝑒 𝑎𝑙𝑙 linear, thus the equation is linear)
+
𝑑𝑦
𝑑𝑥
− 𝑠𝑖𝑛𝑦 = −𝑥
( the term sin y is non-linear, thus the equation is not linear)

8. Linearity – Summary
y
2
dt
dy
dt
dy
y
dt
dy
t
2






dt
dy
y
t)
sin
3
2
(  y
y )
3
2
( 2

Linear Non-linear
2
y )
sin( y
or

9. Thanks!
Any questions?
You can find me at:
+ lory.bulay-og@ustp.edu.ph
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