differential equations engineering mathematics
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May 08, 2024
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About This Presentation
Engineering Mathematics for GATE
Slide Content
DifferentialEquations
Differential
Equations
DifferentialEquations
Differential
Equations
DifferentialEquations
DifferentialEquations
❖Teaching Experience: 10+ Years
❖Qualification: M.Tech from NIT Bhopal
❖Achievements:
▪Cracked GATE Many Times with Good GATE Score & rank
▪Written Many Research and Review Papers
❖Area of Expertise:
▪Strength of Materials, Theory of Machines, Machine Design,
Industrial Engineering & Production.
DHEERAJ SARDANA
ABOUT ME
❖Teaching Experience: 10+ Years
❖Qualification: M.Tech from NIT Bhopal
❖Achievements:
▪Cracked GATE Many Times with Good GATE Score & rank
▪Written Many Research and Review Papers
❖Area of Expertise:
▪Strength of Materials, Theory of Machines, Machine Design,
Industrial Engineering & Production.
DHEERAJ SARDANA
ABOUT ME
DifferentialEquations
DifferentialEquations
DifferentialEquations
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Find Link in the Description.
DifferentialEquations
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Find Link in the Description.
DifferentialEquations
DifferentialEquations
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SARDANA
Join My Telegram Group
https://t.me/dheeraj_sardana
✓For daily dose to improve concepts
✓For daily important questions
✓For latest information about exams
✓For notification of the class
✓For important PDFs & notes
✓For clearing the doubts by experts
DHEERAJ
SARDANA
DifferentialEquations
DifferentialEquations
Find Link in the Description.
Find Link in the Description.
DifferentialEquations
Order & degree of Differential equation:
The order of highest ordered derivative occurring in D. E. is
known as Order of D.E
Degree of a D.E. is highest ordered derivative occurring in it
when derivatives are free from fractional powers.
Order & degree of Differential equation:
The order of highest ordered derivative occurring in D. E. is
known as Order of D.E
Degree of a D.E. is highest ordered derivative occurring in it
when derivatives are free from fractional powers.
DifferentialEquations
DifferentialEquations
Q. The partial differential equation
�
2
∅
��
2
+
�
2
∅
��
2
+
�∅
��
+
�∅
��
=0 has
(a)Degree 1 order 2 (b) degree 1 order 1
(c) Degree 2 order 1 (d) degree 2 order 2
Q. The partial differential equation
�
2
∅
��
2
+
�
2
∅
��
2
+
�∅
��
+
�∅
��
=0 has
(a)Degree 1 order 2 (b) degree 1 order 1
(c) Degree 2 order 1 (d) degree 2 order 2
DifferentialEquations
Solution of Ordinary Differential Equations
First Method
Variable separable
(I)
��
��
=
��
��
→�����??????�� �� � ����
→�����??????�� �� � ����
or
��
��
=��∙��
Solution of Ordinary Differential Equations
First Method
Variable separable
(I)
��
��
=
��
��
→�����??????�� �� � ����
→�����??????�� �� � ����
or
��
��
=��∙��
DifferentialEquations
Q. The solution of the differential
equation
��
��
+�
2
=0 is
(a) y=
1
�+�
(b) �=
−�
3
3
+�
(c) c�
�
(d) Unsolvable as equation is non-linear
Q. The solution of the differential
equation
��
��
+�
2
=0 is
(a) y=
1
�+�
(b) �=
−�
3
3
+�
(c) c�
�
(d) Unsolvable as equation is non-linear
DifferentialEquations
DifferentialEquations
Solution of Ordinary Differential Equations
(II)
��
��
=���+��+�
Solution of Ordinary Differential Equations
(II)
��
��
=���+��+�
DifferentialEquations
Q.
��
��
=�+�+1
2
Q.
��
��
=�+�+1
2
DifferentialEquations
DifferentialEquations
Q.
�??????
�??????
− 2??????=0 is applicable over −10<�<10
if ??????4=10 then ??????−5 is
Q.
�??????
�??????
− 2??????=0 is applicable over −10<�<10
if ??????4=10 then ??????−5 is
DifferentialEquations
DifferentialEquations
Q. for D.E.
��
�??????
+5�=0 �0=1 find solution.
A. �
5??????
B. �
−5??????
C. 5�
−5??????
D. �
−5??????
Q. for D.E.
��
�??????
+5�=0 �0=1 find solution.
A. �
5??????
B. �
−5??????
C. 5�
−5??????
D. �
−5??????
DifferentialEquations
DifferentialEquations
Q. If at every point of a certain curve the slope of the
tangent
−2�
�
the curve is
A. straight line
B. parabola
C. circle
D. ellipse
Q. If at every point of a certain curve the slope of the
tangent
−2�
�
the curve is
A. straight line
B. parabola
C. circle
D. ellipse
DifferentialEquations
DifferentialEquations
Q. Solution of first order D.E.
ሶ��=−3�� �0=�
0 is
A. ��=�
0 �
−3??????
B. ��=�
0 �
−3
C. ��=�
0 �
−??????/3
D. ��=�
0 �
−??????
Q. Solution of first order D.E.
ሶ��=−3�� �0=�
0 is
A. ��=�
0 �
−3??????
B. ��=�
0 �
−3
C. ��=�
0 �
−??????/3
D. ��=�
0 �
−??????
DifferentialEquations
DifferentialEquations
Q. The solution of
��
��
=�
2
with initial value �0=1
bounded in this interval is
A. −∞≤�≤∞
B. −∞≤�≤1
C. �<1,�>1
D. −2≤�≤2
Q. The solution of
��
��
=�
2
with initial value �0=1
bounded in this interval is
A. −∞≤�≤∞
B. −∞≤�≤1
C. �<1,�>1
D. −2≤�≤2
DifferentialEquations
DifferentialEquations
Q. The solution of differential equation
��
��
−�
2
=1
satisfying condition �0=1 is
A. �=�
�
2
B. �=cot�+
??????
4
C. �=�
D. �=tan�+
??????
4
Q. The solution of differential equation
��
��
−�
2
=1
satisfying condition �0=1 is
A. �=�
�
2
B. �=cot�+
??????
4
C. �=�
D. �=tan�+
??????
4
DifferentialEquations
DifferentialEquations
Q. Match the following
P:
��
��
=
�
�
Q:
��
��
=−
�
�
R:
��
��
=
�
�
S:
��
��
=−
�
�
(1) Circle
(2) St. line
(3) Hyperbola
Q. Match the following
P:
��
��
=
�
�
Q:
��
��
=−
�
�
R:
��
��
=
�
�
S:
��
��
=−
�
�
(1) Circle
(2) St. line
(3) Hyperbola
DifferentialEquations
DifferentialEquations
Q. The general solution of
��
��
=cos�+� with � as
constant
A. �+sin(�+�)
B. cos
�+�
2
=�+�
C. tan
�+�
2
=�+�
D. tan
�+�
2
=�+�
Q. The general solution of
��
��
=cos�+� with � as
constant
A. �+sin(�+�)
B. cos
�+�
2
=�+�
C. tan
�+�
2
=�+�
D. tan
�+�
2
=�+�
DifferentialEquations
DifferentialEquations
Q. Which one of the following is the general solution
of 1
st
order D.E.
��
��
=�+�−1
2
; � & � are real
A. �=1+�+tan
−1
(�+�) � is constant
B. �=1+�+tan�+� � is constant
C. �=1−�+tan
−1
�+� � is constant
D. �=1−�+tan�+� � is constant
Q. Which one of the following is the general solution
of 1
st
order D.E.
��
��
=�+�−1
2
; � & � are real
A. �=1+�+tan
−1
(�+�) � is constant
B. �=1+�+tan�+� � is constant
C. �=1−�+tan
−1
�+� � is constant
D. �=1−�+tan�+� � is constant
DifferentialEquations
DifferentialEquations
Linear Differential EQN
(I)
��
��
+ �
�=� � & � = FNS of � only
Linear Differential EQN
(I)
��
��
+ �
�=� � & � = FNS of � only
DifferentialEquations
DifferentialEquations
Linear Differential EQN
(II)
��
��
+ �
�=� � & � = FNS of � only
Linear Differential EQN
(II)
��
��
+ �
�=� � & � = FNS of � only
DifferentialEquations
DifferentialEquations
Q. If �
2
∙
��
��
+2��=
2ln�
�
�1=0 find �(�)
A. �
B. 1
C. 1/�
D. 1/�
2
Q. If �
2
∙
��
��
+2��=
2ln�
�
�1=0 find �(�)
A. �
B. 1
C. 1/�
D. 1/�
2
DifferentialEquations
DifferentialEquations
Q.
��
��
+2��=�
−�
2
with �0=1
A.1+��
�
2
B.1+��
−�
2
C.1−��
�
2
D.1−� �
−�2
Q.
��
��
+2��=�
−�
2
with �0=1
A.1+��
�
2
B.1+��
−�
2
C.1−��
�
2
D.1−� �
−�2
DifferentialEquations
DifferentialEquations
Q. The solution of the differential equation
��
��
+2��=�
(−�
2
)
with y(0) = 1 is
(a) (1 + x) �
(+�
2
)
(b) (1 + x) �
(−�
2
)
(c) (1 - x) �
(+�
2
)
(d (1 - x) �
(−�
2
)
Q. The solution of the differential equation
��
��
+2��=�
(−�
2
)
with y(0) = 1 is
(a) (1 + x) �
(+�
2
)
(b) (1 + x) �
(−�
2
)
(c) (1 - x) �
(+�
2
)
(d (1 - x) �
(−�
2
)
DifferentialEquations
DifferentialEquations
Q. �.
��
��
+�=�
4
�1=
6
5
A. �=
�
4
5
+
1
�
B. �=
4�
4
5
+
4
5�
C. �=
�
4
5
+1
D. �=
�
5
5
+1
Q. �.
��
��
+�=�
4
�1=
6
5
A. �=
�
4
5
+
1
�
B. �=
4�
4
5
+
4
5�
C. �=
�
4
5
+1
D. �=
�
5
5
+1
DifferentialEquations
DifferentialEquations
Homogeneous Differential Equations :-
��
��
=
�(�,�)
��,�
→both are homogeneous
→function
��,�=�
3
+�
3
+�
2
�→ power of each term
is same
Condition
���,��=�
�
��,�⇒ Homogeneous
���,��≠�
�
��,�⇒ Non-Homogeneous
Homogeneous Differential Equations :-
��
��
=
�(�,�)
��,�
→both are homogeneous
→function
��,�=�
3
+�
3
+�
2
�→ power of each term
is same
Condition
���,��=�
�
��,�⇒ Homogeneous
���,��≠�
�
��,�⇒ Non-Homogeneous
DifferentialEquations
DifferentialEquations
Q. A curve passing through a point1,
??????
6
Let the
slope of curve at each point (�,�) be
�
�
+sec
�
�
.
Then eqn. of curve �>0
A. sin
�
�
=ln�+
1
2
B. cos
�
�
=log�+
1
2
C. sin
2�
�
=log�+2
D. cos
2�
�
=log�+
1
2
Q. A curve passing through a point1,
??????
6
Let the
slope of curve at each point (�,�) be
�
�
+sec
�
�
.
Then eqn. of curve �>0
A. sin
�
�
=ln�+
1
2
B. cos
�
�
=log�+
1
2
C. sin
2�
�
=log�+2
D. cos
2�
�
=log�+
1
2
DifferentialEquations
DifferentialEquations
Second order linear diff. Eqn :
�
�
�
��
�
+
��
�−1
�
��
�−1
+
��
�−2
�
��
�−2
+⋯+��=��
� & � are constant
for second order �=2
�
2
�
��
2
+
���
��
+��=��
Second order linear diff. Eqn :
�
�
�
��
�
+
��
�−1
�
��
�−1
+
��
�−2
�
��
�−2
+⋯+��=��
� & � are constant
for second order �=2
�
2
�
��
2
+
���
��
+��=��
DifferentialEquations
DifferentialEquations
Homogeneous linear DE
�
2
�
��
2
+
���
��
+�
�=0
� & � are constant
Homogeneous linear DE
�
2
�
��
2
+
���
��
+�
�=0
� & � are constant
DifferentialEquations
DifferentialEquations
Q.
�
2
�
��
2
−5
��
��
+6=0
Q.
�
2
�
��
2
−5
��
��
+6=0
DifferentialEquations
DifferentialEquations
Case 2 :
If roots are real & equal
C.F. = �
1+�
2� �
??????�
four roots are equal
�.??????.=�
1+�
2�+�
3�
2
+�
4�
3
�
??????�
Case 2 :
If roots are real & equal
C.F. = �
1+�
2� �
??????�
four roots are equal
�.??????.=�
1+�
2�+�
3�
2
+�
4�
3
�
??????�
DifferentialEquations
DifferentialEquations
Case 3 :
If roots are complex
�=�±??????�
�=�.??????.=�
��
�
1cos��+�
2sin��
Case 3 :
If roots are complex
�=�±??????�
�=�.??????.=�
��
�
1cos��+�
2sin��
DifferentialEquations
DifferentialEquations
�
2
�
��
2
+
��
��
+�=0
�
2
�
��
2
+
��
��
+�=0
DifferentialEquations
DifferentialEquations
Q. Given an ordinary D.E.
�
2
�
��
2
+
��
��
−6�=0 �0=0
��
��
0=1
Find value of �(1) :
Q. Given an ordinary D.E.
�
2
�
��
2
+
��
��
−6�=0 �0=0
��
��
0=1
Find value of �(1) :
DifferentialEquations
DifferentialEquations
Q. The maximum value of solution �� of D.E.
��+�″(�)=0 with initial condition �
′
0=1
& �0=1 for �≥0 is
A. 1
B. 2
C. ??????
D.2
Q. The maximum value of solution �� of D.E.
��+�″(�)=0 with initial condition �
′
0=1
& �0=1 for �≥0 is
A. 1
B. 2
C. ??????
D.2
DifferentialEquations
DifferentialEquations
Q. A function �(�) ssatisfies the differential eqn
�
2
��
��
2
−
��
??????
2
=0 where � is a constant
The boundary conditions are �0=� & �∞=0
The soln to eqn is -
A. ��=−� �
−�/??????
B. ��=� �
−�/??????
C. ��=� �
−�/??????
D. ��=�
2
�
−�/??????
Q. A function �(�) ssatisfies the differential eqn
�
2
��
��
2
−
��
??????
2
=0 where � is a constant
The boundary conditions are �0=� & �∞=0
The soln to eqn is -
A. ��=−� �
−�/??????
B. ��=� �
−�/??????
C. ��=� �
−�/??????
D. ��=�
2
�
−�/??????
DifferentialEquations
DifferentialEquations
Particular Integral :
�
2
�
��
2
+
���
��
+�=��
�(�) is a function of � only
→��=�
��+�
→ ��=�
�
→ ��=sin(��+�) or cos(��+�)
Particular Integral :
�
2
�
��
2
+
���
��
+�=��
�(�) is a function of � only
→��=�
��+�
→ ��=�
�
→ ��=sin(��+�) or cos(��+�)
DifferentialEquations
DifferentialEquations
Case 1:
��=�
��+�
�
2
�
��
2
+
2��
��
+6�=�
2�+1
Case 1:
��=�
��+�
�
2
�
��
2
+
2��
��
+6�=�
2�+1
DifferentialEquations
DifferentialEquations
�
2
�
��
2
−
2��
��
+�=�
�
�
2
�
��
2
−
2��
��
+�=�
�
DifferentialEquations
DifferentialEquations
Case 2:
��=sin(��+�)
or
cos(��+�)
�
2
�
��
2
+
2��
��
+�=sin2�+1 or cos(2�+1)
Case 2:
��=sin(��+�)
or
cos(��+�)
�
2
�
��
2
+
2��
��
+�=sin2�+1 or cos(2�+1)
DifferentialEquations
DifferentialEquations
Case 3:
�
2
�
��
2
+
���
��
+��=�
�
→ algebraic function
�
2
�+� ��+��=�
�
Case 3:
�
2
�
��
2
+
���
��
+��=�
�
→ algebraic function
�
2
�+� ��+��=�
�
DifferentialEquations
DifferentialEquations
Q. The solution of D.E.
�
2
�
��
2
+
6��
��
+9�=9�+6
�
1 & �
2 = constants
A.�
1�+�
2 �
−3�
B.�
1�+�
2�
−3�
+�
C. �
1 �
3�
+�
2 �
−3�
D.�
1�+�
2�
3�
+�
Q. The solution of D.E.
�
2
�
��
2
+
6��
��
+9�=9�+6
�
1 & �
2 = constants
A.�
1�+�
2 �
−3�
B.�
1�+�
2�
−3�
+�
C. �
1 �
3�
+�
2 �
−3�
D.�
1�+�
2�
3�
+�
DifferentialEquations
DifferentialEquations
DifferentialEquations
DifferentialEquations
Given that ሷ�+3�=0,��� �0=1,ሶ�0=0, what is
x(1)?
(a)-0.99 (b) -0.16
(c) 0.16 (d) 0.99
Given that ሷ�+3�=0,��� �0=1,ሶ�0=0, what is
x(1)?
(a)-0.99 (b) -0.16
(c) 0.16 (d) 0.99
DifferentialEquations
DifferentialEquations
Q. Consider the differential equation
��
��
=1+�
2
�. The
general solution with constant c is
(a)y=tan
�
2
2
+tan� (b) �=���
2
�
2
+�
(c) �=���
2
�
2
+c (d) y=tan
�
2
2
+�
Q. Consider the differential equation
��
��
=1+�
2
�. The
general solution with constant c is
(a)y=tan
�
2
2
+tan� (b) �=���
2
�
2
+�
(c) �=���
2
�
2
+c (d) y=tan
�
2
2
+�
DifferentialEquations
DifferentialEquations
Consider the differential equation �
2
�
2
�
��
2
+�
��
��
−4�=0
with the boundary conditions of y(0) = 0 and y(1) = 1
The complete solution of the differential equation is
(a)�
2
(b) sin
??????�
2
(c) �
�
sin
??????�
2
(d) �
−�
sin
??????�
2
Consider the differential equation �
2
�
2
�
��
2
+�
��
��
−4�=0
with the boundary conditions of y(0) = 0 and y(1) = 1
The complete solution of the differential equation is
(a)�
2
(b) sin
??????�
2
(c) �
�
sin
??????�
2
(d) �
−�
sin
??????�
2
DifferentialEquations
DifferentialEquations
Find the solution of
�
2
�
��
2
=� which passes
through the origin and the point (ln2,
3
4
)
(a)�=
1
2
�
�
- �
−�
(b)�=
1
2
(�
�
+�
−�
)
(c)�=
1
2
(�
�
−�
−�
)
(d)�=
1
2
�
�
+�
−�
Find the solution of
�
2
�
��
2
=� which passes
through the origin and the point (ln2,
3
4
)
(a)�=
1
2
�
�
- �
−�
(b)�=
1
2
(�
�
+�
−�
)
(c)�=
1
2
(�
�
−�
−�
)
(d)�=
1
2
�
�
+�
−�
DifferentialEquations
DifferentialEquations
For the equation,
��
��
+7�
2
�=0, if y(0)=
3
7
, then the
value of y(1) is
(a)
3
7
�
−
7
3 (b)
7
3
�
−
7
3
(c)
3
7
�
−
3
7 (d)
7
3
�
−
3
7
For the equation,
��
��
+7�
2
�=0, if y(0)=
3
7
, then the
value of y(1) is
(a)
3
7
�
−
7
3 (b)
7
3
�
−
7
3
(c)
3
7
�
−
3
7 (d)
7
3
�
−
3
7
DifferentialEquations
DifferentialEquations
The differential equation
��
��
+4�=5 is valid in
the domain 0≤�≤1 with y(0) =2.25. Then
solution of differential equation is
(a)�= �
−4�
+1.25
(b)�= �
−4�
+5
(c)�= �
4�
+5
(d)�= �
4�
+1.25
The differential equation
��
��
+4�=5 is valid in
the domain 0≤�≤1 with y(0) =2.25. Then
solution of differential equation is
(a)�= �
−4�
+1.25
(b)�= �
−4�
+5
(c)�= �
4�
+5
(d)�= �
4�
+1.25
DifferentialEquations
DifferentialEquations
A differential equation is given as
�
2
�
2
�
��
2
−2�
��
��
+2�=4
The solution of differential equation in terms
of arbitrary constant �
1 ��� �
2 is
(a)�=
??????1
�
2
+ �
2x +2
(b)�=�
1�
2
+ �
2x +4
(c)�=�
1�
2
+ �
2x +2
(d)�=
??????1
�
2
+ �
2x +4
A differential equation is given as
�
2
�
2
�
��
2
−2�
��
��
+2�=4
The solution of differential equation in terms
of arbitrary constant �
1 ��� �
2 is
(a)�=
??????1
�
2
+ �
2x +2
(b)�=�
1�
2
+ �
2x +4
(c)�=�
1�
2
+ �
2x +2
(d)�=
??????1
�
2
+ �
2x +4
DifferentialEquations
DifferentialEquations
DifferentialEquations
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Join My Telegram Group
https://t.me/dheeraj_sardana
✓For daily dose to improve concepts
✓For daily important questions
✓For latest information about exams
✓For notification of the class
✓For important PDFs & notes
✓For clearing the doubts by experts
DHEERAJ
SARDANA
DifferentialEquations
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