Introduction to First Order

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Lecture 10

• Introduction to First Order
• Differential Equations

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Lecture 10 Objectives

• Show that a given function is a solution of a
  given first order DE with/without initial
  condition.
• Given the general solution for a first order DE,
  find the particular solution satisfying a given
  initial condition.
• Find frames of first order DEs.
• Classify DEs according to linearity and order.
• Solve a given first order DE with/without initial
  condition using
• The Fundamental Theorem of Calculus
• Separation of variables.
• Solve a given first order linear DE with/without
  initial condition using the method of integrating
  factors.

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Introduction to Differential Equations

• Definition
• A differential equation (DE) in the variables x
  and y (as independent and dependent variable) is
  an equation involving x and y as well as the
  derivatives y?, y?, y(3), etc..
• The order of a DE is the order of the highest
  derivative in the equation.
• A solution of a DE is a function y g(x)
  ( y(x)) satisfying the equation.

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Examples Find the order of each of the following
DEs

• The differential equation Order
• x3y? 2ysinx
• 1
• y? yy?
• 2
• y(4) y x3
• 4
• (y?)2 2 y? ? 8y 0
• 1
• (x2 y2)dx (x2 ? y2)dy 0
• 1

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Example Check that the function y e3x is a
solution for the DE y? 3y.

• Can you find other solutions?
• Note We can show that solutions of the above DE
  take the form y Ce3x, where C is an arbitrary
  constant. Such a solution is called the general
  solution of the DE.
• This is similar to the case of a general
  anti-derivative (an indefinite integral).

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Example Check that the function y x 1
Cex is always a solution for the DE y? y ?
x.Then find the particular solution satisfying
the initial condition y(0) 2/3.
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Picture
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First Order Differential Equations

• We now seek the general solutions of first order
  DE.
• Usually we can solve the first order DE for y?
  dy/dx to get the following

Note The function f(x,y) is called the frame of
the DE.
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Solving First Order Differential Equations

• Special Case 1 y? f(x) (a function of x only)
• Method Just integrate both sides indefinitely
  (or definitely using the Fundamental Theorem of
  Calculus)

Example Find the general solution of the DE
y? xsinx. Then find the particular solution
satisfying y(0) 1
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Separable First Order DEs

• Special Case 2 y? dy/dx g(x)H(y)
• Method Separate x and dx on one side from y and
  dy on the other, then perform an indefinite
  integration.

Example Find the general solution of the DE
y? (sin2x)(cos2y).
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Separable First Order DEs
Example Find the general solution of the DE
y? ay, where a is a constant. Then
find the particular solution satisfying y(0)
b
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Linear First Order DEs

• Special Case 3 y? P(x)y Q(x)
• Method We multiply both sides by the factor
  v(x) e?P(x)dx.
• Noting that v?(x) v(x)P(x), we can write the
  equation as v(x)y? v(x)Q(x)
• Now, we integrate both sides to get v(x)y
  ? v(x)Q(x)dx

Example Solve the DE y? x y/x
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Applications Electric Circuits
Example The current in the circuit below is
modeled by Ldi/dt Ri V. Find the current as
a function of time if i(0) 0, i.e. the switch
is closed at t 0.
Also, find the steady state current, i.e.
limt?? i(t)
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Graph
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Lecture 10 Objectives (revisited)

• Show that a given function is a solution of a
  given first order DE with/without initial
  condition.
• Given the general solution for a first order DE,
  find the particular solution satisfying a given
  initial condition.
• Find frames of first order DEs.
• Classify DEs according to linearity and order.
• Solve a given first order DE with/without initial
  condition using
• The Fundamental Theorem of Calculus
• Separation of variables.
• Solve a given first order linear DE with/without
  initial condition using the method of integrating
  factors.

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• Thank you for listening.
• Wafik