Merced College; Don Power

Calculus Lecture - Ch 9

9.1 First-Order Differential Equations and Applications

State the order of a DE: first page of lesson

Confirm a given solution to a given DE: by substitution:

Def: Solution to a DE is a function (not a numerical value) that makes the DE true

To test a proposed solution, calculate the required derivatives and plug into the DE

Ex: verify that y = 4cos(3t) - 5sin(3t) is a solution of the 2nd order DE y'' = -9y

Ex: for what values of r does the function y = e^rt satisfy the same DE y'' = -9y ?

[solve r²e^rt = -9e^rt to get r = ±3i]

Confirm that the functions in a given family are solutions to a given DE #3-4

Ex from above: verify that y = C₁cos(3t) + C₂sin(3t) is a solution of the 2nd order DE y'' = -9y

Use implicit diff to confirm a given solution to a given DE: #5-6

Solve DE using integrating factors, apply initial conditions

Solve 1st order linear DE by method of integrating factors: #7-14

Technique applies to first order linear DE, of the form y' + p(x) y = q(x)

Integrating factor is μ = e ^ int(p(x)) μ =

Technique: Calculate the integrating factor;

Omit the "+C" at this step, since any antiderivative works as an integrating factor

Then write y ∙ μ = int( q(x) ∙ μ )

Result is y ∙ μ = f(x) + C Exponential forms: If f(x) = e^g(x), this becomes y ∙ μ = Ce^g(x)

Solve for y: y = 1/μ ∙ f(x) + 1/μ ∙ C For the exponential form, we get y = 1/μ ∙ Ce^g(x)

After getting this result, apply the initial condition to find and replace the value of C

Solve separable equations, finding general solution: #15-24

To be separable, you must be able to write the DE in the forms

dy/dx = f(x) ∙ g(y) result: g(y) dy = f(x) dx ...or

dy/dx = f(x) / g(y) result: dy / g(y) = f(x) dx

Then integrate both sides, putting the "+C" on only the right side

Solve for y if possible, to express the family of solutions as an explicit function of x

After getting this result, apply the initial condition to find and replace the value of C

Sketch several typical integral curves

Apps in later half of the asgn

Geometric (example 6)

Mixing problems (Example 7)

Free fall retarded by air resistance (Discussion in text, solved in X47)

9.2 Direction Fields; Euler's Method OMIT

Lecture:

Slopefields

Manual technique: Ex, y' = x+y

Maple: with(DEtools);

dfieldplot(diff(y(x),x)=x+y(x),[y(x)],x=-3..3,y=-3..3)

TI-89: Mode DE; y1' = y1+t

TI-86: Mode DE; Q1' = Q1+t

Graphmatica: dy = y+x {0,1} -- to specify an initial condition

Show how to estimate the graph of a solution with a given initial condition

DEs for electronics: Explain the DEs for

RL circuit: E = L * dI/dL + RI

RC circuit: E = RI + Q/C, but since I = dQ/dt:

E = R * dQ/dt +Q/C

Note: Eqn for RL circuit becomes E = L * d²Q/dt² + R * dQ/dt

and RLC circuit becomes E = L * Q'' + R * Q' + 1/C * Q

Hence, Second degree DE's are common in electronics

Omit the numerical approach for Euler's method

9.3 Modeling with First-Order Differential Equations

Fact: Modeling real-world phenomena often involves observing how fast quantities change;

since rates of change are derivatives, the models are often differential equations

Def: a DE is an eqn that contains an unknown function and its derivatives

Classic example: unconstrained population growth:

model: growth is proportional to the population

mathematically, dy/dt = ky, where k is the constant of proportionality (the growth rate)

Sol of basic exp growth eqn dy/dt = ky yields y(t) = y₀e^kt

Note: k = growth rate

"Relative growth rate": k = y' / y

Growth: bacteria

Doubling time vs half-life

Radioactive decay

Half-life: show that in general, k = -ln(2) / halflife

Show how to set up eqn to calculate half-life

Continuous compound interest

Basic comp int formula

Take limit as n goes to ¥

A = Pe^rt note r = interest rate

Example of constrained population growth (e.g. limited food supply, spread of disease)

dy/dt = ky(L-y), where L is the maximum sustainable level for the population

Note that there is no growth if y is 0 or L,

max growth if y is halfway between 0 and L

neg growth if y > L (N/A for disease in a population of L indiv)

Other Examples

by Hooke's Law, the force on a spring is proportional to the extension of the spring

F = -kx (negative because the force is opposite to the direction of the extension)

Since (Newton) F = ma and a = second derivative, we get

m * d²x/dt² = -kx

This is a second order DE (order of a DE is the level of the highest derivative in the eqn, here, 2)

Analysis of y'=y(y-2): note techniques:

Constant solutions will be where y' = 0. Solve by factoring, in this case

y is increasing when y' >0 [you're now solving a polynomial inequality]

y is decreasing when y' <0

9.4 Second-Order Linear Homogeneous Differential Equations; The Vibrating Spring OMIT

Additional Notes

Orthogonal trajectories:

Ex: Graphs: y=ke^-x for several values of k; y=±sqrt(2x+c) for several values of c

Method: y'=-ke^-x; eliminate k: y' = -y; integrate neg recip y' = 1/y

If you do not eliminate k, you get a family that is orthogonal to a single graph (with a fixed k):

y=ke^-x; orthogonal graphs y=(1/k)e^x+c for several values of c and a fixed k=1)

Method: y'=-e^-x; integrate neg recip y' = e^x

Trig Identity Note

a cos t + b sin t = sqrt(a^2+b^2) sin (t + arctan(a/b))

Proof: Let a/sqrt(a^2+b^2) and b/ sqrt(a^2+b^2) = sinq and cosq, for rt triangle with legs a and b.

Mult and divide the expression a cos t + b sin t by sqrt(a^2+b^2) and sub in sinq and cosq.

Use identity for sin(q + t)

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