DIFFERENTIAL EQUATION-DEFINITION AND CLASSIFICATION OF D.E-SOLUTION OF D.E Problems

DIFFERENTIAL EQUATION INSTRUCTIONAL MATERIALS
PROBLEM SETS
TOPIC: DEFINITION AND CLASSIFICATION OF D.E.
SOLUTION OF D.E.

DEFINITION AND CLASSIFICATION OF D.E.
; SOLUTION OF D.E.
PROBLEM SET




EASY

6𝑦 2
1. y’=
6𝑥



𝑑𝑦 6𝑦 2
=
𝑑𝑥 6𝑥

𝑑𝑦 𝑑𝑦 𝑑𝑦
∫ 6𝑦2= ∫ 6𝑥 𝑑𝑥

1 𝑑𝑦 1 𝑑𝑥
∫
6 𝑦2
= 6∫ 𝑥

1 𝑦 −1 −1
( )=( 𝑦 𝑑𝑦)
6 −1

1 1
− = 𝑙𝑛𝑥 + 𝑐
6𝑦 6

−y=lnx+c GENERAL SOLUTION
𝑑𝑦
2. 𝑑𝑥
= 4𝑥 3 𝑒 −𝑦

x=1

y=1

𝑑𝑦
∫ 𝑒 −𝑦 =∫ 4𝑥 3 𝑑𝑥


∫ 𝑒 𝑦 𝑑𝑦 = 4 ∫ 𝑥 3 𝑑𝑥


(𝑒 𝑦 = 𝑥 4 + 𝑐)𝑙𝑛


𝑙𝑛𝑒 𝑦 =ln𝑥 4 +c


𝒚 = 𝐥𝐧(𝒙𝟒 ) + 𝒄 GENERAL SOLUTION

DEPARTMENT OF COMPUTER ENGINEERING

, DIFFERENTIAL EQUATION INSTRUCTIONAL MATERIALS
PROBLEM SETS
TOPIC: DEFINITION AND CLASSIFICATION OF D.E.
SOLUTION OF D.E.

1=ln(1)4 +c


1=0+c


1=c


y=ln𝒙𝟒 + 𝟏 PARTICULAR SOLUTION

𝑑𝑦
3. (𝑥 2 + 1) = 𝑥𝑦
𝑑𝑥


𝑑𝑦
((𝑥 2 + 1) = 𝑥𝑦) ((𝑥 2 + 1)𝑦)
𝑑𝑥

𝑑𝑦 𝑥
= 2 𝑑𝑥
𝑦 (𝑥 + 1)

𝑑𝑦 𝑥
∫ =∫ 2 𝑑𝑥
𝑦 (𝑥 + 1)

𝑥
ln 𝑦 = ∫ 𝑑𝑥
(𝑥 2 + 1)

Let u = (𝑥 2 + 1)

𝑑𝑢 = 2𝑥𝑑𝑥

𝑑𝑢
= 𝑥𝑑𝑥
2

1 𝑑𝑢
ln 𝑦 = ∫
2 𝑢

1
ln 𝑦 = ln(𝑢) + 𝐶
2
1
ln 𝑦 = ln(𝑥 2 + 1) + 𝐶
2
1 2 +1)
𝑒 𝑙𝑛𝑦 = 𝑒 2 ln(𝑥 ∙ 𝑒𝐶

𝒚 = 𝐥𝐧 √(𝒙𝟐 + 𝟏) + 𝑪 GENERAL SOLUTION

4. 𝑚𝑦𝑑𝑥 = 𝑛𝑥𝑑𝑦

𝑚𝑑𝑥 𝑛𝑑𝑦
∫ =∫
𝑥 𝑦


DEPARTMENT OF COMPUTER ENGINEERING

, DIFFERENTIAL EQUATION INSTRUCTIONAL MATERIALS
PROBLEM SETS
TOPIC: DEFINITION AND CLASSIFICATION OF D.E.
SOLUTION OF D.E.

𝑑𝑥 𝑑𝑦
𝑚∫ = 𝑛∫
𝑥 𝑦

𝑚 ln(𝑥) = 𝑛𝑙𝑛(𝑦) + 𝐶
𝑚 𝑛
𝑒 ln(𝑥) = 𝑒 ln(𝑦) ∙ 𝑒 𝐶

𝒙𝒎 = 𝒚𝒏 + 𝑪 GENERAL SOLUTION

𝑑𝑦
5. 𝑑𝑥
= 2𝑥 sin 3𝑥

𝑑𝑦 = (2𝑥 + sin 3𝑥)𝑑𝑥

∫ 𝑑𝑦 = ∫(2𝑥 + sin 3𝑥)𝑑𝑥

2𝑥 2 1
𝑦= − cos 3𝑥 + 𝑐
2 3
𝟏
𝒚 = 𝒙𝟐 − 𝟑 𝐜𝐨𝐬 𝟑𝒙 + 𝒄 GENERAL SOLUTION

𝑑𝑦 13
6. (𝑦 2 − 1) = 3𝑦 given that 𝑦 = 1 when 𝑥 =
𝑑𝑥 6

𝑦2 − 1
𝑑𝑥 = 𝑑𝑦
3𝑦

𝑦2 − 1
∫ 𝑑𝑥 = ∫ 𝑑𝑦
3𝑦

𝑦 1
∫ 𝑑𝑥 = ∫( − )𝑑𝑦
3 3𝑦

𝒚𝟐 𝟏
𝒙= 𝟔
− 𝟑 𝐥𝐧 𝒚 + 𝑪 GENERAL SOLUTION

13
When 𝑥 = 6
,𝑦 = 1

13 12 1
= − ln(1) + 𝐶
6 6 3
13 1 1
𝑐= − − ln(1) =
6 6 3

𝑐=2

𝒚𝟐 𝟏
𝒙= 𝟔
− 𝟑 𝐥𝐧 𝒚 + 𝟐 PARTICULAR SOLUTION

𝑑𝑦
7. 2𝑥 = 1 + 𝑦2 when 𝑥 = 2 , 𝑦 = 3
𝑑𝑥

DEPARTMENT OF COMPUTER ENGINEERING