20094em3 Exercise 14

Engineering Mathematics 3

FIRST ORDER DIFFERETIAL EQUATIOS

Linear Equation Using Integrating Factor Grade 50% off the total score within the first week after due date. Zero score after the first week. Name : Group: Subject: Topic: Date Due: Date of Submission:

Section A (rhs = 0) Question 1.4.1 Solve y' - 5y = 0 .

− 5x

y⋅ e

Ans: Question 1.4.2 1 Q=0 . Solve Q' + 20

1 20

=k

x

Ans:

y⋅ e

Ans:

y⋅ exp− t +

=k

Question 1.4.3 dy + ( t − 1) ⋅ y = 0 . Solve dt

 

1 2 ⋅t  = K 2 

Section B (rhs is nonzero) Question 1.4.4 Solve y' − 3y = 6 at y(1) = 0. Ans:

y⋅ exp( −3⋅ x) = − 2⋅ exp( −3⋅ x) + 2⋅ exp( − 3)

Ans:

v⋅ exp( 25⋅ t) = .3920⋅ exp( 25.⋅ t) + 56.13

Question 1.4.5 Solve v' + 25⋅ v = 9.8 at v(0.1) = 5.

Question 1.4.6 3 Q=2 . Solve Q' + 100 − t

Q

Ans:

( 100 − t) 20094em3 Exercise 14.mcd

Copyright 2008 - 2009

1 3

= ( 100 − t)

2

+k

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Engineering Mathematics 3

Question 1.4.7 2 Q=4 . Solve Q' + 10 + 2t 2

Q⋅ ( 10 + 2⋅ t) = 40⋅ t + 4⋅ t + k

Ans: Question 1.4.8 dy 3 − 2y = x ⋅ cos( 4x) . Solve x⋅ dx Remark: De is not in the linear standard form. Conversion is necessary. Ans

Question 1.4.9 2 2 4 Solve x' − x = t . t 3

y=

1  2  ⋅ sin( 4⋅ x) + k ⋅ x 4 

; x(t): x is a function of t.

x = 54⋅

Ans

 1 13  ⋅ t + k   1053  t

Question 1.4.10 2 2 Solve y' − y = x − 1 x

at y(1) = 1.

 

y = x +

Ans

Question 1.4.11 Solve I' + 10I = 10

8

1 x

 

2

− 1 ⋅ x

at I(0) = 0.

Question 1.4.12 dy 7 3 = x − 4x ⋅ y. Solve dx

Remark: De is not in the standard linear form. Conversion is necessary. Ans

y=

1 4

(

4

( ) − exp( x ) + 4⋅ k) ⋅ exp( −x ) 4

4

⋅ x ⋅ exp x

4

Question 1.4.13 dy 3 = cos( x) − y⋅ tan( x) . Solve dx

Remark: Convert to linear form. Ans:

20094em3 Exercise 14.mcd

y=

1 2

2

⋅ cos( x) ⋅ sin( x) +

Copyright 2008 - 2009

1 2

⋅ cos( x) ⋅ x + cos( x) ⋅ k

2/3

Engineering Mathematics 3

Question 1.4.14 dy 2 − x⋅ y = 1 . Solve x + 1 dx

(

)

1

(

2

y = x + k⋅ x + 1

Ans:

)

2

Section C (Bernoullian De)

Question 1.4.15 4

Solve y' + 2⋅ x⋅ y + x⋅ y = 0. −1

1

Ans:

3

y

=

2

( ) 2

+ exp 3⋅ x ⋅ k

Question 1.4.16 2

Solve y' + y = y ( cos( x) − sin( x) ) . ( − exp( − x) ⋅ sin( x) + k)

1

Ans:

y

=

exp( − x)

Question 1.4.17 Solve y' + x⋅ y = 6⋅ x⋅ y at y(0) = 0. 1

Ans:

20094em3 Exercise 14.mcd

Copyright 2008 - 2009

y

2

= 6+

1

 1 2 exp ⋅ x  4 

⋅k

3/3