MAT211: Engineering Mathematics
Prepared by
Mizanur Rahman
Lecturer in Mathematics
Department of GED, FSIT,
DIU
Differential Equations
Formation of Differential Equations
Definition: DE, Classification, ODE, PDE, degree, order, solution, General, particular, singular
01
solution, Formation of Differential equations.
Problems:
𝑑2𝑦
)
𝑑𝑥 2
1.
Write down the order and degree of 𝑥 2 (
2.
Is the ODE
3.
Show that the ODE of any straight line is
3
4
4
𝑑𝑦 4
𝑑𝑥
+ 𝑦 ( ) + 𝑦4 = 0 .
 d3y 
d2y
 dy 
2
 3  − 5 x 2 + y = 5   + y − x linear?
dx
 dx 
 dx 
2
𝑑2 𝑦
𝑑𝑥 2
=0.
Eliminate the constant ‘a’ from √1 − 𝑥 2 + √1 − 𝑦 2 = 𝑎(𝑥 − 𝑦) .
5. Derive the differential equation of which 𝑐(𝑦 + 𝑐)2 = 𝑥 3 .
6. Find the differential equation of the family of curves 𝑦 = 𝑒 −𝑥 (𝐴𝑐𝑜𝑠𝑥 + 𝐵𝑠𝑖𝑛𝑥)
where A and B are arbitrary constants.
7. Obtain the ODE associated with the primitive = 𝑎𝑥 2 + 𝑏𝑥 .
4.
8.
Find the differential equation of the family of curves y = Ae2 x + Be−2 x , for
different values of A and B.
9.
From the differential equation of which y = ae x + be− x + c cos x + d sin x is a
solution.
10. Find the differential equation of the family of curves 𝑦 = 𝑒 𝑥 (𝐴𝑐𝑜𝑠𝑥 + 𝐵𝑠𝑖𝑛𝑥)
where A and B are arbitrary constants.
11. Find the differential equation of the solution xy = Ae + Be
x
−x
+ x2 .
First order First degree ODE
Definition: Separation of variable, integrating factor, linear ODE, Bernoulli’s Equation,
02
Homogeneous and Exact ODE.
Problems:
Variable Separable form:
1.
Solve the differential equation
( 4 + y ) dx + ( 4 + x ) dy = 0 by variable separable
2
2
Method.
dy
= 1 − x 2 1 − y 2 by variable separable Method.
dx
2.
Solve the differential equation
3.
Solve the ODE sec x tan ydx + sec y tan xdy = 0 .
4.
Solve
2
2
3
dy
= e x+ y + x 2 e x + y .
dx
Reducible to Variable Separable form:
dy
= sin ( x + y ) + cos ( x + y ) .
dx
dy
= tan ( x + y + 6 ) by choosing appropriate
6. Solve the differential equation
dx
5.
Solve the differential equation
transformation.
7.
Solve
dy
= 1− x + y .
dx
Homogeneous ODE:
dy y
 y
= + tan   .
dx x
x
8.
Solve
9.
dy x 2 + y 2
=
Solve
dx
2 xy
.
10. State the type of the differential equation for the equation xdy − ydx =
solve it.
Non-homogeneous ODE:
11. Solve
( 6 x − 5 y + 4) dy + ( y − 2 x −1) dx = 0 .
12. Solve the differential equation ( 3 x − 2 y + 1)
dy
= 6x − 4 y + 3
dx
Linear ODE:
13. Solve x
dy
+ 2 y = x 2 log x .
dx
14. Solve the differential equation
dy
− 2 y cos x = −2sin 2 x .
dx
x 2 + y 2 dx and
Bernoulli’s ODE:
dy
= x3 y 3 − xy .
dx
dy y
16. Solve the differential equation
+ = y2 .
dx x
15. Solve the differential equation
Exact ODE:
17. Solve
( 2 x + y − 1) dy − ( x − 2 y + 5) dx = 0
18. Solve (𝑥 2 − 4𝑥𝑦 − 2𝑦 2 )𝑑𝑥 + (𝑦 2 − 4𝑥𝑦 − 2𝑥 2 )𝑑𝑦 = 0
Linear ODE with first degree and higher order with
constant coefficients
03
Definition: Auxiliary Equation, Particular integral, complementary function and
formula for finding particular integral.
Problems:
2
1. Solve D y − 3Dy + 2 y = 0
d3y
dy
−3 + 2y = 0
3
dx
dx
2
d y
dy
− 2 + 5y = 0
3. Solve
2
dx
dx
4
2
4. Solve D y − 4 D y + 4 y = 0
2.
Solve
5.
Solve D y − 81y = 0
6.
Solve D 2 y + 5Dy + 4 y = 3 − 2 x .
7.
Solve D 2 y − 6 Dy + 9 y = 1 + x + x 2 .
8.
Solve D 2 y + 4 Dy + 4 y = e 2 x + e −2 x
9.
Solve D y − 8Dy + 16 y = 5cos3x
4
2
10. Solve
D3 y − 2 Dy + 4 y = e x cos x .
Linear ODE with first degree and higher order with
variable coefficients
04
Problems:
d2y
dy
+ 9 x + 25 y = 0
2
dx
dx
2
dy
2 d y
− 2 x − 4 y = x4
2. Solve x
2
dx
dx
2
dy
2 d y
+ 4x + 2 y = ex
3. Solve x
2
dx
dx
1.
Solve x 2
d2y
dy
− x − 3y = 0
2
dx
dx
2
dy
2 d y
5. Solve x
− x − 3 y = x 2 ln x
2
dx
dx
2
dy
2 d y
6. Solve x
+ 4 x + 2 y = x + sin x
2
dx
dx
4.
Solve x 2
Laplace Transformation
Definition: Introduction, Laplace transform, properties and related proofs, Hyperbolic
01
sine & cosine functions.
Problems:
Derive the followings:
1.
L 1 =
1
s
2.
L t =
1
s2
3.
L t n  =
4.
L e a t  =
5.
L sinat =
a
s + a2
6.
L cosat =
s
s + a2
7.
L sinh at =
s
s + a2
8.
L coshat =
s
s − a2
n !  ( n + 1)
=
s n +1
s n +1
1
s−a
2
2
2
2
Properties Related Problems :
Linearity property:


L e4t + 4t 3 − 2sin 3t + 3cos5t
1.
𝓛{4𝑒
2.
5𝑡
3
+ 6𝑡 − 3𝑠𝑖𝑛𝑡 + 2𝑐𝑜𝑠2𝑡}
First shifting property:
1. 𝓛{𝑒 2𝑡 (𝑠𝑖𝑛𝑡 + 3𝑐𝑜𝑠3𝑡}

L e−t sin 2 4t
2.

n
Multiplication by t :
1.
2.
Show that L t sin 2t = ?

2
3. Show that

3
4. Show that

Show that L t e
5t
=?


L t 2 cos t =
2s 3 − 6s
(s
2
)
+1
3
L t cos 2t = ?
Division by t:
1.
 e −t sin t 

 t

 cos at − cos bt 
L

t


2. L 
Inverse Laplace Transformation
Definition: Inverse Laplace transform and its properties.
02
Problems:
Linearity property:
1.
2.


2s 2 − 4
L−1 

( s + 1)( s − 2)( s − 3) 
s


L −1  2

 s + 4s + 3 
−1


3s + 1

2
( s − 1)( s + 1) 
3. L 
First Shifting Property:
1.
 s+5 
L −1  2

 s + 4s + 5 
2. L
−1
s
 4s + 3 

−1 
 2
 3. L  2

 s + 9 s + 25 
 s + 6s + 5 
Second Shifting Property:
1.
 e − s 
L  2

 s + 5s + 9 
−1
 e − s

2. L  2

 s + 6s + 5 
−1
Multiplication by tn / Inverse Laplace transform of derivative
1.
1

L −1  tan −1 
s

2. L
−1
cot ( s + 1)
−1
Division by t/ Inverse Laplace transform of integrals
1.
2.
L
−1
L
−1


1
3
u  
  2

+ 3+ 2
+ 2
  2
 du 

s  u + 4 u u − 2 u + 4  




5


du 

2

 s (u + 1)(u + 5u + 6) 

Convolution theorem:
−1


1

( s − 1)( s − 2) 
1.
Find L 
2.
Find ℒ −1 {𝑠 2(𝑠+1)2}
1
𝑠
3. Find ℒ −1 {(𝑠 2+𝑎2)2}
1
4. Find ℒ −1 {(𝑠+1)(𝑠2+1)}
Application:
(a) Solve IVP Y + Y = t ,
''
(b)
(c)
(d)
(e)
Y (0) = 1, Y '(0) = −2
Solve the IVP 𝑌"(𝑡) + 9𝑌 = 40𝑒 𝑡 , 𝑌(0) = 5, 𝑌′(0) = −2
Solve the IVP 𝑌 ′′ + 𝑌 = 𝑡, 𝑌(0) = 1, 𝑌 ′(0) = −2
Solve the IVP 𝑌 ′′ − 3𝑌 ′ + 2𝑌 = 4𝑒 2𝑡 , 𝑌 (0) = −3, 𝑌 ′ (0) = 5
Solve the IVP 𝑌 ′′ + 𝑌 = 8 cos 𝑡 , 𝑌 (0) = 1, 𝑌 ′ (0) = −1
Fourier Transformation
Definition: Fourier transform, Fourier sine & cosine transform.
Finite Fourier sine / cosine Transformation
01
(a) F ( x) = 2 x , 0  x  4
, 0 x 
(c) F ( x) = sin mx , 0  x  
(d) F ( x) = cos mx , 0  x  
(b) F ( x) = e
mx
Infinite Fourier sine / cosine Transformation
(a) F ( x) = e
− ax
, 0 x
(b) F ( x) = e
−2 x
+ e−5 x , 0  x  
Application:
Use Finite Fourier Transform to solve the boundary value problem
U  2U
= 2 ;U (0, t ) = U (4, t ) = 0, U ( x, 0) = 2 x where 0  x  4, t  0 .
t
x
Fourier Series
Definition: Fourier series, Even, Odd functions, half range sine and cosine
series.
01
Problems:
Full range Fourier Series
(a) F ( x) = x + x 2 , −  x  
x2
(b) F ( x) = x +
, −  x  
4
(c) F ( x) = sin ax , -  x  
Half range Fourier Series
(a) F ( x) = eax , 0  x  
(b) F ( x) = x , 0  x  2
End