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Differential Equation

A differential equation is an equation involving derivatives of an unknown function and the independent variable. There are ordinary and partial differential equations. The order of a differential equation is the highest order of the derivative in the equation. Examples of common differential equations include equations for free falling objects, springs, Newton's law of cooling, and growth and decay models.

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Allan Añavisa Ostique Jr.
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61 views4 pages

Differential Equation

A differential equation is an equation involving derivatives of an unknown function and the independent variable. There are ordinary and partial differential equations. The order of a differential equation is the highest order of the derivative in the equation. Examples of common differential equations include equations for free falling objects, springs, Newton's law of cooling, and growth and decay models.

Uploaded by

Allan Añavisa Ostique Jr.
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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DIFFERENTIAL EQUATION

A differential equation is an equation involving derivatives of an


unknown function and possibly the function itself as well as the
independent variable.

Example
4
y ′ = sin ( x ) , ( y ') − y 2 + 2 xy − x 2 = 0, y ′′ + y 3 + x = 0

1st order equations 2nd order equation

y is dependent variable and x is independent variable,


and these are ordinary differential equations.

The order of a differential equation is the highest order of the


derivatives of the unknown function appearing in the equation

Example

y ′ = sin ( x ) ⇒ y = − cos ( x ) + C

y ′′ = 6 x + e x ⇒ y ′ = 3 x 2 + e x + C1 ⇒ y = x 3 + e x + C1x + C2
Niel Arvin B. Galos
Example

∂ 2u ∂ 2u
2
+ 2 =0
∂x ∂y
u is dependent variable and x and y are independent
variables, and is partial differential equation.

A differential equation is linear, if


1. dependent variable and its derivatives are of degree one,
2. coefficients of a term does not depend upon dependent
variable.

Example
4
2
d y dy d 3 y  dy 
+ 3 + 9 y = 0. 3
+   + 6y = 3
dx 2
dx dx  dx 

is linear. is non - linear because in 2nd term is not of


degree one.
Niel Arvin B. Galos
1. Free falling stone

d 2s
2
= −g
dt
where : s is distance or height and
g is acceleration due to gravity.

2. Spring vertical displacement

d2y
m 2 = −ky
dt
where: y is displacement,
m is mass and
k is spring constant

Niel Arvin B. Galos


3. Newton’s Law of Cooling

dT
= κ (T − Ts )
dt
dT
where : is rate of cooling of the liquid,
dt

T − Ts is temperature difference between the


liquid ‘T’ and its surrounding Ts

4. Growth and Decay


dy
=κ y
dt
y is the quantity present at any time

Niel Arvin B. Galos

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