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Section 1.1 Differential Equations & Mathematical Models. Differential Equations – “Equations with derivatives in them.” Examples: 1. 2. 3. y ″ – 2 = 3 x ln ( x ) + y 2. What is a solution to a differential equation?.

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Section 1 1 differential equations mathematical models
Section 1.1 Differential Equations & Mathematical Models


Section 1 1 differential equations mathematical models

Differential Equations – “Equations with derivatives in them.”

Examples:

1.

2.

3. y″ – 2 = 3xln(x) + y2



Section 1 1 differential equations mathematical models

A differential equation will often have infinitely many solutions.

For example, here are some of the many solutions of

1. y = x2

2. y = x2 + e–x

3. y = x2 + 4e–x

4. y = x2 + 31429674e–x

:

:

“Family of solutions”


Section 1 1 differential equations mathematical models

Calculus Review: solutions.

If then


Section 1 1 differential equations mathematical models

A differential equation will usually have infinitely many solutions, but there times when a differential equation will have only one solution or no solutions.

Example:

(y′)2 + y2 = –1


Section 1 1 differential equations mathematical models

Some differential equations will have solutions, but unfortunately we can't write them down (in terms of our elementary functions).

Example:


Section 1 1 differential equations mathematical models

General Solutions unfortunately we can't write them down (in terms of our elementary functions). vs. Particular Solutions

1. Solve

2. Solve if y = 5 when x = 1.

3. Solve , y(1) = 5.


Section 1 1 differential equations mathematical models

Initial Value Problem – unfortunately we can't write them down (in terms of our elementary functions).

consists of a differential equation along with an initial condition y(xo) = yo


Section 1 1 differential equations mathematical models

Definition: Order unfortunately we can't write them down (in terms of our elementary functions).

The order of a differential equation is the order of the highest derivative appearing in it.


Section 1 1 differential equations mathematical models

Expressing differential equations: unfortunately we can't write them down (in terms of our elementary functions).

Often we will be able to express 1st order differential equations as


Section 1 1 differential equations mathematical models

Expressing differential equations: unfortunately we can't write them down (in terms of our elementary functions).

We will always be able to express. . . .

1st order differential equations in the form F(x, y, y′) = 0

2nd order differential equations in the form F(x, y, y′, y″) = 0

:

nth order differential equations in the form F(x, y, y′, y″, y″′, . . . . , y(n)) = 0


Section 1 1 differential equations mathematical models

Definition: Solution to a Differential Equation unfortunately we can't write them down (in terms of our elementary functions).

A function u(x) is a solution to the differential equation F(x, y, y′, y″, . . , y(n)) = 0 on an interval J if u, u′, u″, . . . , u(n) exist on J and F(x, u, u′, u″, . . . , u(n)) = 0 for all x on J.


Section 1 1 differential equations mathematical models

Ex. 1 unfortunately we can't write them down (in terms of our elementary functions).

(a) Show that y(x) = 1/x is a solution to on the interval [1, 20].


Section 1 1 differential equations mathematical models

Ex. 1 unfortunately we can't write them down (in terms of our elementary functions).

(b) Show that y(x) = 1/x is not a solution to on the interval [-20, 20].


Section 1 1 differential equations mathematical models

Ex. 2 unfortunately we can't write them down (in terms of our elementary functions).

(a) Show that y1(x) = sin(x) is a solution to (y′ )2 + y2 = 1

(b) Show that y2(x) = cos(x) is a solution to (y′ )2 + y2 = 1


Section 1 1 differential equations mathematical models

Partial Derivatives unfortunately we can't write them down (in terms of our elementary functions).

Ordinary Differential Equations vs. Partial Differential Equations


Section 1 2 integrals as general particular solutions
Section unfortunately we can't write them down (in terms of our elementary functions).1.2 Integrals as General & Particular Solutions


Section 1 1 differential equations mathematical models

Ex. 1 unfortunately we can't write them down (in terms of our elementary functions). Solve


Section 1 1 differential equations mathematical models

Ex. 2 unfortunately we can't write them down (in terms of our elementary functions). Solve


Section 1 1 differential equations mathematical models

Ex. 3 unfortunately we can't write them down (in terms of our elementary functions). Solve


Section 1 1 differential equations mathematical models

Position - Velocity – Acceleration unfortunately we can't write them down (in terms of our elementary functions).

s(t) = position s′ (t) = velocity s″ (t) = acceleration

Force = Mass x Acceleration


Section 1 1 differential equations mathematical models

Ex. 5 unfortunately we can't write them down (in terms of our elementary functions). A lunar lander is falling freely toward the surface of the moon at a speed of 450 m/s. Its retrorockets, when fired, provide a constant deceleration of 2.5 m/s2 (the gravitational acceleration produced by the moon is assumed to be included in the given deceleration). At what height above the lunar surface should the retro rockets be activated to ensure a "soft touchdown" (velocity = 0 at impact)?


Section 1 1 differential equations mathematical models

Ex. 5 unfortunately we can't write them down (in terms of our elementary functions). A lunar lander is falling freely toward the surface of the moon at a speed of 450 m/s. Its retrorockets, when fired, provide a constant deceleration of 2.5 m/s2 (the gravitational acceleration produced by the moon is assumed to be included in the given deceleration). At what height above the lunar surface should the retro rockets be activated to ensure a "soft touchdown" (velocity = 0 at impact)?


Section 1 3 slope fields solution curves
Section unfortunately we can't write them down (in terms of our elementary functions).1.3 Slope Fields & Solution Curves


Section 1 1 differential equations mathematical models

Slope field for unfortunately we can't write them down (in terms of our elementary functions).


Section 1 1 differential equations mathematical models

Slope field for unfortunately we can't write them down (in terms of our elementary functions).


Section 1 1 differential equations mathematical models

Ex. 1 unfortunately we can't write them down (in terms of our elementary functions). Sketch the slope field for y′ = –x


Section 1 1 differential equations mathematical models

Ex. 2 unfortunately we can't write them down (in terms of our elementary functions). Sketch the slope field for y′ = x2 + y2


Section 1 1 differential equations mathematical models

Ex. 3 unfortunately we can't write them down (in terms of our elementary functions). Examine some solution curves of

On the following slope field, draw the solution curve which satisfies the initial condition of. . . . .

(a)y(2) = –1

(b)y(–1) = 3

(c)y(0) = 0

(d)y(0) = 1


Section 1 1 differential equations mathematical models

Calculus Review unfortunately we can't write them down (in terms of our elementary functions).(definition of continuity):

f (x) is continuous at xo if


Section 1 1 differential equations mathematical models

Calculus Review unfortunately we can't write them down (in terms of our elementary functions).(definition of continuity):

f (x) is continuous at xo if

f (x, y) is continuous at (xo, yo) if


Section 1 1 differential equations mathematical models

Theorem I: Existence & Uniqueness of Solutions unfortunately we can't write them down (in terms of our elementary functions).

Suppose that f (x, y) is continuous on some rectangle in the xy-plane containing the point (xo, yo) in its interior and that the partial derivative fy is continuous on that rectangle. Then the initial value problem has a unique solution on some open interval Jo containing the point xo.


Section 1 1 differential equations mathematical models

Ex. 4 unfortunately we can't write them down (in terms of our elementary functions). Determine what this theorem says about the solutions in each of the following differential equations:

(a)


Section 1 1 differential equations mathematical models

Ex. 4 unfortunately we can't write them down (in terms of our elementary functions). Determine what this theorem says about the solutions in each of the following differential equations:

(b)


Section 1 1 differential equations mathematical models

Ex. 4 unfortunately we can't write them down (in terms of our elementary functions). Determine what this theorem says about the solutions in each of the following differential equations:

(c)


Section 1 1 differential equations mathematical models

Ex. 4 unfortunately we can't write them down (in terms of our elementary functions). Determine what this theorem says about the solutions in each of the following differential equations:

(d)


Section 1 1 differential equations mathematical models

Ex. 4 unfortunately we can't write them down (in terms of our elementary functions). Determine what this theorem says about the solutions in each of the following differential equations:

(e)


Section 1 1 differential equations mathematical models

Ex. 4 unfortunately we can't write them down (in terms of our elementary functions). Determine what this theorem says about the solutions in each of the following differential equations:

(f)


Section 1 4 separable equations applications
Section unfortunately we can't write them down (in terms of our elementary functions).1.4 Separable Equations & Applications


Section 1 1 differential equations mathematical models

Definition: Separable Differential Equation unfortunately we can't write them down (in terms of our elementary functions).

A first order differential equation is said to be separable if

f (x, y) can be written as a product of a function of x and a function of y

(i.e. ).


Section 1 1 differential equations mathematical models

Definition: Separable Differential Equation unfortunately we can't write them down (in terms of our elementary functions).

A first order differential equation is said to be separable if

f (x, y) can be written as a product of a function of x and a function of y

(i.e. ).

Examples:

1.

2.

3.



Section 1 1 differential equations mathematical models

To solve a separable differentiable equation of the form

we proceed as follows:

h(y) dy = g(x) dx


Section 1 1 differential equations mathematical models

To solve a separable differentiable equation of the form

we proceed as follows:

h(y) dy = g(x) dx

(Then integrate both sides and solve for y, if this is possible.)


Section 1 1 differential equations mathematical models

Ex. 1 Solve


Section 1 1 differential equations mathematical models

Ex. 2 Solve


Section 1 1 differential equations mathematical models

Justification for why this method for solving separable differentiable equations actually works.


Section 1 1 differential equations mathematical models

Ex. 3 differentiable equations actually works. Solve


Section 1 1 differential equations mathematical models

Ex. 4 differentiable equations actually works. Solve


Section 1 1 differential equations mathematical models

Review of general solutions and particular solutions. differentiable equations actually works.


Section 1 1 differential equations mathematical models

Definition: Singular Solution differentiable equations actually works.

A particular solution to a first order differential equation is said to be a singular solution if it does not come from the general solution.


Section 1 1 differential equations mathematical models

Ex. 5 differentiable equations actually works. Solve


Section 1 1 differential equations mathematical models

Ex. 5 differentiable equations actually works. Solve


Section 1 1 differential equations mathematical models

In general: differentiable equations actually works.

If we have the differential equation , and h(y) has a zero of yo

then the function y(x) = yo will be a singular solution.


Section 1 1 differential equations mathematical models

Applications differentiable equations actually works.

If y changes at a rate proportional to y then (for some constant k).

Radioactive material & half-lives


Section 1 1 differential equations mathematical models

Ex. 6 differentiable equations actually works. A radioactive substance has a half-life of 5 years. Initially there are 128 grams of this substance. How much remains after t years?


Section 1 1 differential equations mathematical models

Ex. 7 differentiable equations actually works. The half-life of radioactive cobalt is 5.27 years. Suppose that a nuclear accident has left the level of cobalt radiation in a certain region at 100 times the level acceptable for human habitation. How long will it be until the region is again habitable?


Section 1 1 differential equations mathematical models

Carbon dating differentiable equations actually works.


Section 1 1 differential equations mathematical models

Ex. 8 differentiable equations actually works. Carbon extracted from an ancient skull contained only one-sixth as much 14C as carbon extracted from present-day bone. How old is the skull?


Section 1 1 differential equations mathematical models

Newton's law of cooling (heating) differentiable equations actually works.

According to Newton's law of cooling, the time rate of change of the temperature T of a body immersed in a medium of constant temperature A is proportional to

the difference T – A . That is:


Section 1 1 differential equations mathematical models

Ex. 9 differentiable equations actually works. A cake is removed from an oven at 210° F and left to cool at a room temperature, which is 70° F. After 30 min the temperature of the cake is 140° F. When will it be 100° F?


Section 1 5 linear first order equations
Section differentiable equations actually works.1.5 Linear First-Order Equations


Section 1 1 differential equations mathematical models

Definition: Linear Differential Equation (First Order) differentiable equations actually works.

A first order differential equation is linear if there are functions P(x) and Q(x) so that

Examples:

1. y′ + sin(x) y =ex

2. y′ – sin(x) y =ex

3. y′ = 3x2y + x3 – 4x + 1

4. cos(x) y′ – sec(x) y = x2


Section 1 1 differential equations mathematical models

Definition: Linear Differential Equation (First Order) differentiable equations actually works.

A first order differential equation is linear if there are functions P(x) and Q(x) so that

Integrating Factor:


Section 1 1 differential equations mathematical models

Steps you MUST show when solving a 1 differentiable equations actually works.st order linear differential equation:

1. Put the differential equation in the form:

2. Compute μ.

3. Multiply μ on both sides of the differential equation to obtain

4. Write this as (μy)′ = μ Q(x)

5. Solve this last differential equation via integration.


Section 1 1 differential equations mathematical models

Ex. 1 differentiable equations actually works. Solve:


Section 1 1 differential equations mathematical models

Why isn't there a “+C” in the integrating factor differentiable equations actually works.μ?


Section 1 1 differential equations mathematical models

Ex. 2 differentiable equations actually works. Solve: x y′ + 2y = 10x3; y(1) = 5


Section 1 1 differential equations mathematical models

Reminder: (Theorem I from section 1.3) differentiable equations actually works.

Suppose that f (x, y) is continuous on some rectangle in the xy-plane containing the point (xo, yo) in its interior and that the partial derivative fyis continuous on that rectangle. Then the initial value problem

has a unique solution on some open interval Jo containing the point xo.


Section 1 1 differential equations mathematical models

Theorem I differentiable equations actually works.

If the functions P(x) and Q(x) are continuous on the open interval J containing the point xo, then the initial value problem

has a unique solution y(x) on J .


Section 1 1 differential equations mathematical models

Ex. 3 differentiable equations actually works. Solve:


Section 1 1 differential equations mathematical models

Reminder of a result from calculus: differentiable equations actually works.


Section 1 1 differential equations mathematical models

Applications differentiable equations actually works.

Mixture Problems

Solutes, Solvents, & Solutions

Q = amount of solute, V = volume of solution

ri = rate in ro = rate out, ci = concentration in, co = concentration out


Section 1 1 differential equations mathematical models

Ex. 4 differentiable equations actually works. Consider a large tank holding 1000 L of water into which a brine solution of salt begins to flow at a constant rate of 6 L/min. The solution inside the tank is kept well stirred and is flowing out of the tank at a rate of 6 L/min. If the concentration of salt in the brine entering the tank is 1 kg/L, determine when the concentration of salt in the tank will reach 0.5 kg/L.


Section 1 1 differential equations mathematical models

Ex. 5 differentiable equations actually works. For the mixture problem described in example 4, assume now that the brine leaves the tank at a rate of 5 L/min instead of 6 L/min and assume that the tank starts out with a concentration of 0.1 kg/L (everything else stays the same as it was in example 4 though). Determine the concentration of salt in the tank as a function of time.


Section 1 1 differential equations mathematical models

Ex. 6 differentiable equations actually works. A swimming pool whose volume is 10,000 gallons contains water that is 0.01% chlorine. Starting at t = 0, city water containing 0.001% chlorine is pumped into the pool at a rate of 5 gal/min, and the pool water flows out at the same rate. What is the percentage of chlorine in the pool after 1 hr? When will the pool be 0.002% chlorine?


Section 1 6 substitution methods exact equations
Section differentiable equations actually works.1.6 Substitution Methods & Exact Equations


Section 1 1 differential equations mathematical models

Ex. 1 differentiable equations actually works. Solve


Section 1 1 differential equations mathematical models

If a differential equation can be written as differentiable equations actually works.

let u = Ax + By + C and the resulting differential equation in terms of u and x will be separable.


Section 1 1 differential equations mathematical models

Definition: First Order Homogenous Differential Equation differentiable equations actually works.

A 1st order differential equation that can be expressed as

is said to be homogenous.

Examples:

1.

2.

3.

4.


Section 1 1 differential equations mathematical models

If a differential equation can be written as let

and the resulting differential equation in terms of u and x will be separable.


Section 1 1 differential equations mathematical models

Ex. 2 let Solve


Section 1 1 differential equations mathematical models

Here's why a 1 let st order homogenous differential equation can be turned into a

separable differential equation by the substitution


Section 1 1 differential equations mathematical models

Here's why a 1 let st order homogenous differential equation can be turned into a

separable differential equation by the substitution


Section 1 1 differential equations mathematical models

Here's why a 1 let st order homogenous differential equation can be turned into a

separable differential equation by the substitution


Section 1 1 differential equations mathematical models

Here's why a 1 let st order homogenous differential equation can be turned into a

separable differential equation by the substitution


Section 1 1 differential equations mathematical models

Here's why a 1 let st order homogenous differential equation can be turned into a

separable differential equation by the substitution


Section 1 1 differential equations mathematical models

Definition: Bernoulli Differential Equation let

A 1st order differential equation that can be expressed as

is said to be a Bernoulli differential equation.

Examples:

1.

2.

3.


Section 1 1 differential equations mathematical models

Ex. 3 let Solve


Section 1 1 differential equations mathematical models

Ex. 3 let Solve


Section 1 1 differential equations mathematical models

Definition: Exact Differential Equation let

A 1st order differential equation that can be expressed as

with is said to be an exact differential equation.


Section 1 1 differential equations mathematical models

Exact Differential Equation let

M(x, y) dx + N(x, y) dy = 0


Section 1 1 differential equations mathematical models

Exact Differential Equation let

Solving M(x, y) dx + N(x, y) dy = 0


Section 1 1 differential equations mathematical models

Ex. 4 let Solve (4x – y) dx + (6y – x) dy = 0


Section 1 1 differential equations mathematical models

Ex. 5 let Solve


Section 1 8 acceleration velocity models
Section let 1.8 Acceleration-Velocity Models


Section 1 1 differential equations mathematical models

Ex. 1 let Suppose that an object is a distance of ho from the surface of the earth when it is given an initial velocity of vo. Determine s(t), the position of the object expressed as a function of time. (Assume that the only force acting on the object is due to gravity. Assume a constant acceleration due to gravity of g).


Section 1 1 differential equations mathematical models

Ex. 2 let Suppose that an object is a distance of ho from the surface of the earth when it is given an initial velocity of vo. Determine s(t), the position of the object expressed as a function of time, this time including air resistance. (Use g for the acceleration due to gravity and assume that the force caused by the air resistance is proportional to the velocity.)


Section 1 1 differential equations mathematical models

Ex. 3 let An object of mass 3 kg is released from rest 500 m above the ground and allowed to fall under the influence of gravity. Assume the force due to air resistance is proportional to the velocity of the object with a drag coefficient of 4/3 sec–1. Determine when the object will strike the ground.


Section 1 1 differential equations mathematical models

Newton's Law of Gravitation – let

The gravitational force of attraction between two point masses M and m

located at a distance r apart is given by


Section 1 1 differential equations mathematical models

Ex. 4 let A lunar lander is falling freely toward the surface of the moon at a speed of 450 m/s. Its retrorockets, when fired, provide a constant deceleration of 2.5 m/s2 . At what height above the lunar surface should the retro rockets be activated to ensure a soft touchdown (velocity = 0 at impact)?