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CONCORDIA UNIVERSITY / Engineering / ENGR 213 / What are differential equations?

# What are differential equations? Description

##### Description: Chapters for the midterm
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"}.lst-kix_o7y7rq4si9gt-3 > li{counter-increment:lst-ctn-kix_o7y7rq4si9gt-3}ol.lst-kix_hb8zkmnd809f-8.start{counter-reset:lst-ctn-kix_hb8zkmnd809f-8 0}.lst-kix_h6i3urvsyts3-3 > li{counter-increment:lst-ctn-kix_h6i3urvsyts3-3}.lst-kix_hb8zkmnd809f-7 > li{counter-increment:lst-ctn-kix_hb8zkmnd809f-7}ol.lst-kix_o7y7rq4si9gt-6.start{counter-reset:lst-ctn-kix_o7y7rq4si9gt-6 0}ol.lst-kix_h6i3urvsyts3-5{list-style-type:none}ol.lst-kix_h6i3urvsyts3-4{list-style-type:none}ol.lst-kix_h6i3urvsyts3-3{list-style-type:none}ol.lst-kix_h6i3urvsyts3-2{list-style-type:none}.lst-kix_h6i3urvsyts3-2 > li:before{content:"" counter(lst-ctn-kix_h6i3urvsyts3-2,lower-roman) ". "}ol.lst-kix_h6i3urvsyts3-8{list-style-type:none}ol.lst-kix_h6i3urvsyts3-7{list-style-type:none}ol.lst-kix_h6i3urvsyts3-6{list-style-type:none}.lst-kix_ojw8ousktcpl-0 > li{counter-increment:lst-ctn-kix_ojw8ousktcpl-0}.lst-kix_h6i3urvsyts3-0 > li:before{content:"" counter(lst-ctn-kix_h6i3urvsyts3-0,decimal) ". 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"}.lst-kix_it5xga617b9d-6 > li:before{content:"- "}.lst-kix_o7y7rq4si9gt-6 > li:before{content:"" counter(lst-ctn-kix_o7y7rq4si9gt-6,decimal) ". "}.lst-kix_ojw8ousktcpl-2 > li:before{content:"" counter(lst-ctn-kix_ojw8ousktcpl-2,lower-roman) ". "}ol.lst-kix_ojw8ousktcpl-3.start{counter-reset:lst-ctn-kix_ojw8ousktcpl-3 0}.lst-kix_o7y7rq4si9gt-4 > li:before{content:"" counter(lst-ctn-kix_o7y7rq4si9gt-4,lower-latin) ". "}.lst-kix_h6i3urvsyts3-4 > li{counter-increment:lst-ctn-kix_h6i3urvsyts3-4}ol.lst-kix_o7y7rq4si9gt-8.start{counter-reset:lst-ctn-kix_o7y7rq4si9gt-8 0}.lst-kix_it5xga617b9d-4 > li:before{content:"- "}ol.lst-kix_h6i3urvsyts3-7.start{counter-reset:lst-ctn-kix_h6i3urvsyts3-7 0}ol.lst-kix_ojw8ousktcpl-2.start{counter-reset:lst-ctn-kix_ojw8ousktcpl-2 0}ol.lst-kix_o7y7rq4si9gt-7.start{counter-reset:lst-ctn-kix_o7y7rq4si9gt-7 0}

Initial Value Problem        (IVP)We also discuss several other topics like What is the equation of homogenous?

1.2 initial value problems

can be considered in many ways

1. As a function, a function of all real numbers X except X=
2. To a solution of the DE , on the interval I.

→ x cant be equal to  because the function cant be defined and differentiate.

1. As a solution of the IVP ; y(0) = -1, on the initial I, (x ) the interval also needs to contain the initial point X=0

Don't forget about the age old question of What are the challenges that arise being a manager?

IVP: its a solution of a DE that satisfies initial conditions

Steps:

1. We find the solution of the DE
2. We plug in the initial conditions to find C()
3. We rewrite the solution for the IVP with the value of C

1.3 Differential equation (DEs) as mathematical modelsWe also discuss several other topics like Describe the functional divisions of the nervous system.

The mathematical description of a system or a phenomenon is growth, radioactivity, heating,....

1. first , wee have to read the situation and identify the variables
2. Make a set of reasonable assumptions about the system

→ they include any applicable laws to the system.If you want to learn more check out What is an atomic number?

→ the assumptions will include a note of change (frequently)

1. Find the mathematical model
2. Solve the model

If the results are weak, we increase the level of resolution:If you want to learn more check out Enumerte the developing stages of a child according to piaget.

Assumptions and hypotheses → express assumptions in terms of DEs → mathematical formation → solve the DE → obtain solution → display predictions of the model(graphically) → check model predictions with known facts → necessary, alter assumption or increase resolution of the model.We also discuss several other topics like Differentiate competitive and noncompetitive inhibitors of enzymes.

• Of course, by increasing the resolution we add to the complexity of the mathematical model and increase the chances of not not finding an explicit solution.
• In a physical system, the variable t will often be involved

→ the dependent variable(s) describe the system in the fast, preset or future.

Population dynamics.

• The more people there is at time t, the more there going to be in the future.
• In mathematical therms: (PLH denotes the total population at time t)

∝P or kP

Where K is a constant of proportionality.

Remark, this model however fail to take things like em/immigration into account

→ good for small population over short intervals (em: bacteria growing)

• The rate DA/Dt at which a substance decays is proportional to the amount A(t) of the substance remaining at the time t:

Decrease → k(-)         ∝A or

This is the equation as in the population growth.

1. A=A(5) → A(0) = C
2. Sample go from 100mg to 68mg

C=100                A=68

In 100 days, it decrease by 33%(66%left)        → 68=100

Conclusion for both equation: → it single differential equation can serve as a mathematical model for many different phenomena.

MM are often accompanied with side conditions.

→ ex: an initial population Po, if this population is at t = 0

Then → P(0) = Po

Mathematical model → initial value problem(condition) IVP

→ boundary - value problem(section 3.9)

Newton`s law of cooling / warming

∝.T-Tm        or =k(T-Tm)

Where T(t): represents the temp of a body at time t

Tm : the temp of the surrounding medium.

: the rate at which the temp. Of the body changes

k : constant of proportionality (k<0 if Tm is constant)

Where  x(t): denotes the number of people having the disease

y(t): number of people who have not been yet exposed.

: the rate of which the disease spreads

→ this rate is proportional to the number of interaction between the two groups (x(t) and y(t))

So it is also proportional to the product of Xy

K: constant of proportionality.

→ em: a community as a amount n of people.

1 person with a disease is introduced in this community.

So → x(t) and y(t) are related by x + y = n + 1

=kx(n + 1 - x)

At exponential growth →

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