Lesson 15: Exponential Growth and Decay (Section 021 slides)Matthew Leingang
Many problems in nature are expressible in terms of a certain differential equation that has a solution in terms of exponential functions. We look at the equation in general and some fun applications, including radioactivity, cooling, and interest.
formulation of first order linear and nonlinear 2nd order differential equationMahaswari Jogia
• Equations which are composed of an unknown function and its derivatives are called differential equations.
• Differential equations play a fundamental role in engineering because many physical phenomena are best formulated mathematically in terms of their rate of change.
• When a function involves one dependent variable, the equation is called an ordinary differential equation (ODE).
• A partial differential equation (PDE) involves two or more independent variables.
Figure 1: CHARACTERIZATION OF DIFFERENTIAL EQUATION
FIRST ORDER DIFFERENTIAL EQUATION:
FIRST ORDER LINEAR AND NON LINEAR EQUATION:
A first order equation includes a first derivative as its highest derivative.
- Linear 1st order ODE:
Where P and Q are functions of x.
TYPES OF LINEAR DIFFERENTIAL EQUATION:
1. Separable Variable
2. Homogeneous Equation
3. Exact Equation
4. Linear Equation
i. SEPARABLE VARIABLE:
The first-order differential equation:
Is called separable provided that f(x,y) can be written as the product of a function of x and a function of y.
Suppose we can write the above equation as
We then say we have “separated” the variables. By taking h(y) to the LHS, the equation becomes:
Integrating, we get the solution as:
Where c is an arbitrary constant.
EXAMPLE 1.
Consider the DE :
Separating the variables, we get
Integrating we get the solution as:
3 examples of PDE, for Laplace, Diffusion of Heat and Wave function. A brief definition of Fouriers Series. Slides created and compiled using LaTeX, beamer package.
Lesson 15: Exponential Growth and Decay (Section 021 slides)Matthew Leingang
Many problems in nature are expressible in terms of a certain differential equation that has a solution in terms of exponential functions. We look at the equation in general and some fun applications, including radioactivity, cooling, and interest.
formulation of first order linear and nonlinear 2nd order differential equationMahaswari Jogia
• Equations which are composed of an unknown function and its derivatives are called differential equations.
• Differential equations play a fundamental role in engineering because many physical phenomena are best formulated mathematically in terms of their rate of change.
• When a function involves one dependent variable, the equation is called an ordinary differential equation (ODE).
• A partial differential equation (PDE) involves two or more independent variables.
Figure 1: CHARACTERIZATION OF DIFFERENTIAL EQUATION
FIRST ORDER DIFFERENTIAL EQUATION:
FIRST ORDER LINEAR AND NON LINEAR EQUATION:
A first order equation includes a first derivative as its highest derivative.
- Linear 1st order ODE:
Where P and Q are functions of x.
TYPES OF LINEAR DIFFERENTIAL EQUATION:
1. Separable Variable
2. Homogeneous Equation
3. Exact Equation
4. Linear Equation
i. SEPARABLE VARIABLE:
The first-order differential equation:
Is called separable provided that f(x,y) can be written as the product of a function of x and a function of y.
Suppose we can write the above equation as
We then say we have “separated” the variables. By taking h(y) to the LHS, the equation becomes:
Integrating, we get the solution as:
Where c is an arbitrary constant.
EXAMPLE 1.
Consider the DE :
Separating the variables, we get
Integrating we get the solution as:
3 examples of PDE, for Laplace, Diffusion of Heat and Wave function. A brief definition of Fouriers Series. Slides created and compiled using LaTeX, beamer package.
This is the entrance exam paper for ISI MSQE Entrance Exam for the year 2004. Much more information on the ISI MSQE Entrance Exam and ISI MSQE Entrance preparation help available on http://crackdse.com
This is the entrance exam paper for ISI MSQE Entrance Exam for the year 2008. Much more information on the ISI MSQE Entrance Exam and ISI MSQE Entrance preparation help available on http://crackdse.com
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
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Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
Biological screening of herbal drugs: Introduction and Need for
Phyto-Pharmacological Screening, New Strategies for evaluating
Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
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Antifertility, Toxicity studies as per OECD guidelines
Francesca Gottschalk - How can education support child empowerment.pptxEduSkills OECD
Francesca Gottschalk from the OECD’s Centre for Educational Research and Innovation presents at the Ask an Expert Webinar: How can education support child empowerment?
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
The French Revolution Class 9 Study Material pdf free download
Mcq differential and ordinary differential equation
1. 17. DIFFERENTIATION AND
ORDINARYDIFFERENTIAL
EQUATION
MULTIPLE CHOICE QUESTIONS
1]. Choose the correct statement from following,
- - -
(a) Every function has a limit
(b) The function having limit is continuous
(c) Every continuous function has limit
(d) The continuity implies differentiability
2. Apartial differential equation has,-
(a) One independent variable
(b) Two or more Independent variables
(c) More than one dependent variable
(d) Equal number ofdependent and independent variables
3J. Degree of differential equation is, - -
(a) Exponent power of highest derivative
(b) Highest derivative in the equation
(c) Lowest derivative in the equation
(d) Exponent power of lowest derivative
41. Orderofdifferentialequation is, -
(a) Lowest derivative in the equation
(b) Highest derivative in the equation
(c)Exponent power of highest derivative
(d) Exponent power of lowest derivative
114
2. and degrecofthedifferentialequation+ +y=x
J O r d e
are
respectively,
-
-
-
(b) 4,2
(a) 1, 2
(d)2,1
(c)
2,4
Theorderofthedifterential equation given by+4y = sinz
(b) 4
(d) 0
(a) 2
(c)1
1.
The steps to obtain a diferential equation are given below. Arrange
them in the correct order,
1) Using the equations obtained, eliminate the arbitrary constants.
2Differentiate the given function w,.r.t the independent variable
present in the equation.
3) Keep differentiating as many times as the number of arbitrary
constants.
Choose the correct order from the options given below.
(a)1,2,3
()2, 1,3
(b)
2,3,1
(d) 1,3, 2
81.Thesolution ofthesecond-order differential equation contains
arbitrary constants.
(a)1
c)3
(b) 2
(d) 0
91.Order and degree ofthe differential equation =0 are
respectively, - - -
(a) 1,2 (b) 3,2
() 2,2 (d)2,1
10).If F = 2x3y +2xy then Fx is, -
(a) 12x +6y
(c) 6xy + 4x
(b)6xy +4xy
(d) 12xy + 4y
1.If F =
3x3y +3xy3 then Fyy is, - - -
(a) 18xy
(c) 3x3 +9x
(b) 9xy
(d) 12x
115
3. ON
121. i s thecondition of- - differentiation.
dx
(a) P'artial (b) Successive
(c) Exact (d) Total
dy
131.Ify = cosx then 1,
(b) -2xsinx2
(d) 2xcosx2
(a) 2xcosx
(c) 2xsinx2
indas y=
fløtz)p
141. Which ofthe following gives chain rule to find
(a) X du (b) yd
( (c)=dx dx du dx
2
15]. Auxiliary equationofthe differentialequation-4
-4+2y=
canbe--
(a) m- 4m +4 =0
(c)m + 4m +4 = 0
(b)
m2 - 4m +2 =0
(d)m+4m +2=0
161. If F = 2x*y + 3xy - 4xy then Ey is---
a)6xy+6x-8
(c)6xy +6x - 8y
(b)6x +6x-8y
(d)6x +6xy -8
17. Which ofthe following is not a type of differential equation?
(a) Ordinary differential equation
(b) Successive differential equation
(c) Linear differential equation
(d) Homogeneous differential equation
18J. Which of the following is a second-order differential cquation?
(a)(y +x=y
(c)y'y"+y=sin x
191.Which ofthe following is the general solution dy
dx
(b) y'= y2
(d) y+x=y
dx
(a) y = (Ax + B)ex
(b) y = Acosx + Bsinx
(c) y = Ae* + Be-*
(d) y =(Ax + B)ex
116
4. 0The
se
(a)
y=e(r-1)
Thesolutionof+yey(0) =0is,,-
(b) y =
xe
(d)y =xe+1
(c)y = x e *
dy
Integrating factor ofthe differential equation+ytanx - secx =
dx
0is,
(b) cosx
(a) secx
(d) e secx
(c)e
os
21.Ify= e(Acosx + sinx), then y is a solution of,-.
6)+2=0
+ 2 y = 0
dy =
0
(a)+2+2y =
0
dx
1 Integrationfactorforthedifferentialequation x=1+2xy is, -.
(a)ex (b) e-2x
(c) ex< (d) e2x
241.1f the general solution of a differential equation is (y + c) = cx,
where c is an arbitrary constant, then the order and degree of
differential equation is, - -
(a) 1,2
(c)2,3
(b)2,1
(d) None of these
25].Integrationfactor forthe differential equation x-2xy=is--
dx
(a)ex (b) e2x
(d)-
261. Which of the following equations is an exact differential equation?
(a)(x+1)dx -xydy =0
(6) xdy + (3x -
2y)dx = 0
(c)xydy-ydx =0 (d)
2xydx + (2+r*)dy = 0
7Theequationy =
cx is general solution of---
(a)y'=
)y2
b)y=
(d)y'= 2x
117
5. the lincar differential
281.Which of the following is a solution of the lincar difta
cquation+ =x2?
(b)
xy=+c
(d)y =+c
(a)xy =+ 3
()xy=+c
29].The solution of a differential equation is y = Cqet* 4 Capz
differential equation is given by-
, the
a-7+ 12y = 0 (b)4-7+7y= 0
dx2
()+7+21y = 0 (d) None ofthese
dx
301. If p and q are the degree and order of the differential quation
(4)+3+=4,then thevalueof2p-3q is
2
dx dx
(b)-7
(a) 7
(c) 3
(d)-3
31.Thedegreeofthedifferential equation(1+ =
dx/
(b) 2
(d) 4
-
(a) 1
()3
32]. Which of the following is a solution of the differential equation
=ey +x3e-y ?
dx
(a)e =
e ++c (b)e =
ex ++c
(c)e =
ex
++c (d)e =
e++c
33.The differential equation 2ydx -
(3y -
2x)dy = 0 is, - - -
(a) Homogeneous and linear but not exact
(b) Exact and linear but not
homogeneous
(c) Exact, homogeneous and linear
(d) Exact and
homogeneous but not linear
341.Theorder of differential equation is
always, - -
(a) Rational number
(b) Whole Number
(c) Negative integer
(d) Positive integer
118
6. he differential equation 2cos(y) dx - xy sin(y )dy = 0 ha
(a)e
as an integrating factor
b) as an integrating factor
c)e as an integrating factor
d)3x as an integrating factor
LAn integration factor ofx+(3x +1)y =xe-lxis,--.
b) 3xe
(d)xe
( a )
x e 3 r
c)xe
2
71.Thedifferentialequation+ +r?+y' is,-..
(a)Homogeneous partial
(b) Non-homogeneous partial
(c) Non-homogeneous exact
(d) Homogeneous exact
38|. Particular integral of (D'-3D +2)y = e3* is,---
(
eSx
13
5x
()4
391. Which ofthe following is the linear diferential equation?
3
(k+6Hi
cos
()+ycosx = sinx
dx
(d) None ofthese
40.An auxiliary equation (A.E.) cannot have - - - roots.
(a) Real and unequal (distinct)
(6) Real and equal
(d) Complex and equal
(c) Complex conjugate
The higher-order partial derivatives i.e. second or third order are
called --partialdifferentials.
(a) Exact b) Total
(C) Successive (d) None ofthese
119
7. fferential
421.Which of the following correctly defincs ordinary
cquations?
variable (say
(a) A differential cquation in which a dependent variaht.
(say 'x)
y) depends on only one independent variable (
variable (say
(b) A differential cquation in which an independent variakl.
y) depends on only onc dependent variable (say 'x') y
(say
(c) A differential equation in whicha dependent variable(e
etc.)
depends on one or more independent variables (say'x',te
say
(d) A differetial equation in which an independent variablefa
y') depends on one or more dependent variables (say'x,t
of an
431. Which one of the following is not a criterion for linearit
ordinary differential equation?
(a) The dependent variable y and its derivatives are offirst
degree
(b) The derivatives of the dependent variable y should be of
second degree
(c) No product terms of y and/or any of its derivatives are present
(d) No transcendental functions ofy and/or its derivatives occur
441.Solutionofthedifferentialequationxy=1+x+y+xyis,..
dx
(a)(y-x)-log (x(1+y))= c
(b) (y +x)- log (x) = c
(clog (x(1 + y) = c
(d) y-«)-log (y(1 +«)=c
2
451.xy +yx* + =0 is a, - - - .
(a) Second order, third degree, and linear differential equation
(b) First order, third degree, linear differential equation
(c) Second order, third degree, and non-linear differential equation
(d) First order, third degree, non-linear differential equation
461. What is the order of the partial differential equation?
ou u = 0?
2u
Oxdy
(b) I
(a) 5
(c) 2 (d) 4
120
8. 17
A
mplementary
function of the differential cquation
481.The
roots
of
fauxiliary equatio. for the differential equation
-3+ 2x = 0 is,---
dt
(a)
Ce +Cze?t
( c ) C e + c 2 e t
dt
(b)cjet+ Ce-2t
(d) -3t e~2t
+2+5 0 are,- -
b) Real and unequal (distinct)
(d) None of these
(a) Real and equal
c Imaginary
491. A particular integral of the differential equation
5+6y = es* is, - - .
dr?
)e
(d);e*
(a)etx
(c)ex
S01. If A and B are constants and auxiliary equation has real and equal
roots m m2 = m, then general solution ofhomogeneous second
order differential equation is - - -
(a) (Ax -
B)e
mx
(c) (Ax + B)emx
(b) (Ax +B)e-2mx
(d) (Ax - B)emx
51.
The given differential equationt3+7y = sinx is,--
(a) Homogeneous
(C) Inhomogeneous
(b) Non-homogeneous
d) Noneofthese
52]. Ordinary differential equation contains,
(a) Only one variable
(6) Two or more independent variable
(c) Zero variables
(d) No independent varible
Iotal differential equation can be solving by homogeneous equation
method and, ---
(a) Inspection method
c) Reduction method
(b) Auxillary equation method
(d) All of these
121