ORDINARY DIFFERENTIAL EQUATION

TRY THIS!

Solve the differential equation below:

1 5 2 + 2 = 10 2
2
3
4 ANSWER:

5
= + 3

ANSWER:


+ sin = 1

ANSWER:

= 4 2 − 7


ANSWER:

SECOND ORDER OF ODE

SECONG ORDER DE

REAL AND DIFFERENT 01
ROOTS

= 1 + 2

02 REAL AND EQUAL
ROOT

= ( + )

IMAGINARY (COMPLEX) 03
ROOTS

= ( cos + sin )

2 change to 2 change to change to 1
2

2 • Real and different roots
answers • Prove by 2 > 4

1 • Real and equal root
answer • Prove by 2 = 4

i • Imaginary (complex) roots
• Prove by 2 < 4

BY USING CALCULATOR

MODE DEG SUBSTITUTE PRESS
3x
EQN 2 A,B,C =

BY USING QUADRATIC FORMULA

− ± 2 − 4
= 2

Q1 Solve the second order
differential equation of

2
2 − − 2 = 0

Step 1 Change to an auxiliary equation
2 − − 2 = 0

Step 2 Determine the value of m

2 − − 2 = 0 Solve using calculator. If the answer
− 2 + 1 = 0 is 2 different numbers, then
1 = 2, 2 = −1
factorize the function manually

Step 3 Substitute m into the right formula
Thus, = 2 + −

Solve the second order
differential equation of

2

Q2 3 2 + 2 − = 0

Step 1 Change to an auxiliary equation
3 2 + 2 − 1 = 0

Step 2 Determine the value of m

3 2 + 2 − 1 = 0

3 − 1 + 1 = 0

1 = 1 , 2 = −1
3

Step 3 Substitute m into the right formula
Thus, = 13 + −

Solve the second order
differential equation of

2

Q3 3 2 + 2 − = 0

Step 1 Change to an auxiliary equation

3 2 + 2 − 1 = 0

Step 2 Determine the value of m
2 2 − 4 + 2 = 0
2 − 2 − 1 = 0

1 = 2= 1

Step 3 Substitute m into the right formula
Thus, = ( + )

Solve the second order
differential equation of

2

Q4 3 2 − 2 + = 0

Step 1 Change to an auxiliary equation
3 2 − 2 + 1 = 0

Step 2 Determine the value of m

= − ± 2−4 Solve using Quadratic
formula because
2 2 < 4

= −(−2)± (−2)2−4(3)(1)

2(3)

= 2± −8 also can be retrieved by using
calculator, meanwhile for , press
6
button SHIFT and =
= 2± 8

6

= 1, = 8

36

Step 3 Substitute m into the right formula

Thus, = 31 ( cos 8 + sin 8 )

6 6

Solve the second order
differential equation of

2

Q5 2 + + 8 = 0

Step 1 Change to an auxiliary equation
2 + + 8 = 0

Step 2 Determine the value of m

= − ± 2−4 Solve using Quadratic
formula because
2 2 < 4

= −(1)± (1)2−4(1)(8) also can be retrieved by using
calculator, meanwhile for , press
2(1)
button SHIFT and =
= −1± −31

2

= −1± 31
2

= − 21, = 31
2

Step 3 Substitute m into the right formula

Thus, = −12 ( cos 31 + sin 31 )
2 2

TRY THIS!

1 Solve the second order
2 differential equation of:

3 2
2 + 8 − 9 = 0

ANSWER:

2
2 2 + 3 + 8 = 0

ANSWER:

2
2 + 6 = 0

4 ANSWER:

2
3 2 + 6 − 9 = 0

ANSWER:

REF ERENCE

Bird, J. (2017). Higher Engineering Mathematics (7th Edition). UK.
Routledge.
Anton, H., Bivens, I. C., & Davis, S. (2005). Calculus (8th ed.).
John Wiley & Sons.
Differential equations [Video]. (n. d.). Khan Academy.
https://www.khanacademy.org
Weisstein, Eric W. "Ordinary Differential Equation."
From MathWorld--A Wolfram Web Resource.
https://mathworld.wolfram.com/OrdinaryDifferentialEquation.html

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