MATH 20063 DIFFERENTIAL EQUATIONS
CHAPTER 1: INTRODUCTION: DEFINITIONS
1.1 Differential Equations
Definition: Any equation that contains a derivative or differential is called a differential
equations.
1.2 Classification of differential equations
(a) The Order of a differential equations is the Order of the highest-ordered
derivative in the equation.
(b) The Degree of a differential equations is the same as the power (or exponent)
to which the derivative of highest order is raised.
(c) The Type of differential equations may be ordinary or partial
1.3 Type of solution to a differential equation
(a) Particular Solution: The solution is said particular if it does not contain any
arbitrary constant.
(b) General Solution: The solution is said to be general if it contains at least one
arbitrary constant.
1.4 Elimination of Arbitrary Constants
Rule: The number of differentiating the given equation is the same as the number of
arbitrary constant to be eliminated.
Example: Eliminate the arbitrary constants.
(1) 𝑦 = 𝐶1 + 𝐶2 𝑒 −4𝑥
Solution: Differentiate the given equation twice, we obtain
𝑦 ′ = −4𝐶2 𝑒 −4𝑥
𝑦 ′′ = 16𝐶2 𝑒 −4𝑥
Eliminating 𝐶2 from the two equations, we get,
𝑦 ′′ + 4𝑦′ = 0 Ans.
, SUPPLEMENTARY EXERCISES 1.4 Eliminate the arbitrary constants
1. 𝑦 = 𝐴𝑒 𝑥 + 𝐵𝑥𝑒 𝑥
2. 𝑦 = 𝑥 2 + 𝐶1 𝑥 + 𝐶2 𝑒 −𝑥
3. 𝑦 = 𝑎(𝑠𝑒𝑐𝑥 + 𝑡𝑎𝑛𝑥)
4. 𝑦 = 𝐶1 + 𝐶2 𝑒 𝑥 + 𝐶3 𝑒 −2𝑥
1.5 Families of Curve
1. Draw the figure
2. Write the general equation of the curve based on the given condition.
3. Perform elimination of arbitrary constant
Example: Find the differential equations of the family of circles with
center on the x-axis.
Solution: The equation of the family of circles with center on the axis is
(𝑥 − ℎ)2 + 𝑦 2 = 𝑟 2 ; where h and r are arbitrary constants
Differentiating the equation twice, we obtain
𝑥 − ℎ + 𝑦𝑦 ′ = 0
1 + 𝑦𝑦 ′′ + (𝑦′)2 = 0 Ans.
SUPPLEMENTARY EXERCISES 1.5
Find the differential equations of the family of curves.
1. Parabolas with axis parallel to the x-axis whose distance between the focus
and the vertex is a. Use: (𝑦 − 𝑘)2 = 4𝑎(𝑥 − ℎ)
2. Straight lines passing through (h, k). Use: 𝑚(𝑥 − ℎ) = 𝑦 − 𝑘
3. Circles with center on the y –axis. Use: (𝑥 − ℎ)2 + (𝑦 − 𝑘)2 = 𝑟 2
𝑥2 𝑦2
4. Hyperbolas. Use: 2
− =1
𝑎 𝑏2
CHAPTER 1: INTRODUCTION: DEFINITIONS
1.1 Differential Equations
Definition: Any equation that contains a derivative or differential is called a differential
equations.
1.2 Classification of differential equations
(a) The Order of a differential equations is the Order of the highest-ordered
derivative in the equation.
(b) The Degree of a differential equations is the same as the power (or exponent)
to which the derivative of highest order is raised.
(c) The Type of differential equations may be ordinary or partial
1.3 Type of solution to a differential equation
(a) Particular Solution: The solution is said particular if it does not contain any
arbitrary constant.
(b) General Solution: The solution is said to be general if it contains at least one
arbitrary constant.
1.4 Elimination of Arbitrary Constants
Rule: The number of differentiating the given equation is the same as the number of
arbitrary constant to be eliminated.
Example: Eliminate the arbitrary constants.
(1) 𝑦 = 𝐶1 + 𝐶2 𝑒 −4𝑥
Solution: Differentiate the given equation twice, we obtain
𝑦 ′ = −4𝐶2 𝑒 −4𝑥
𝑦 ′′ = 16𝐶2 𝑒 −4𝑥
Eliminating 𝐶2 from the two equations, we get,
𝑦 ′′ + 4𝑦′ = 0 Ans.
, SUPPLEMENTARY EXERCISES 1.4 Eliminate the arbitrary constants
1. 𝑦 = 𝐴𝑒 𝑥 + 𝐵𝑥𝑒 𝑥
2. 𝑦 = 𝑥 2 + 𝐶1 𝑥 + 𝐶2 𝑒 −𝑥
3. 𝑦 = 𝑎(𝑠𝑒𝑐𝑥 + 𝑡𝑎𝑛𝑥)
4. 𝑦 = 𝐶1 + 𝐶2 𝑒 𝑥 + 𝐶3 𝑒 −2𝑥
1.5 Families of Curve
1. Draw the figure
2. Write the general equation of the curve based on the given condition.
3. Perform elimination of arbitrary constant
Example: Find the differential equations of the family of circles with
center on the x-axis.
Solution: The equation of the family of circles with center on the axis is
(𝑥 − ℎ)2 + 𝑦 2 = 𝑟 2 ; where h and r are arbitrary constants
Differentiating the equation twice, we obtain
𝑥 − ℎ + 𝑦𝑦 ′ = 0
1 + 𝑦𝑦 ′′ + (𝑦′)2 = 0 Ans.
SUPPLEMENTARY EXERCISES 1.5
Find the differential equations of the family of curves.
1. Parabolas with axis parallel to the x-axis whose distance between the focus
and the vertex is a. Use: (𝑦 − 𝑘)2 = 4𝑎(𝑥 − ℎ)
2. Straight lines passing through (h, k). Use: 𝑚(𝑥 − ℎ) = 𝑦 − 𝑘
3. Circles with center on the y –axis. Use: (𝑥 − ℎ)2 + (𝑦 − 𝑘)2 = 𝑟 2
𝑥2 𝑦2
4. Hyperbolas. Use: 2
− =1
𝑎 𝑏2
