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Elimination of aarbitrary constants and family of curves

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Title: Elimination of aarbitrary constants and family of curves


1
Week 2 Elimination of Arbitrary constants Family
o f Curves
  • EMATH 09
  • DIFFERENTIAL EQUATION

2
SOLUTIONS OF D.E
  • 1. General Solution- involving 1 or more
    arbitrary constant
  • Ex y(t)C1ekt C2 ekt
  • 2. Particular Solution- no arbitrary constant
  • Ex p 3.9ekt
  • 3. Complete Solution- combination of two
    solutions (particular and a complimentary
    solution)
  • YYpYc
  • 4. Computer Solution- using computer software

3
ELIMINATION OF ARBITRARY CONSTANTS
  • Methods for the elimination of arbitrary
    constants vary with the way in which the
    constants enter the given relation.
  • A method that is efficient for one problem may
    be poor for another. One fact persists
    throughout. Because each differentiation yields a
    new relation, the number of derivatives that need
    be used is the same as the number of arbitrary
    constants to be eliminated. We shall in each case
    determine the differential equation that is
  • (a) Of order equal to the number of arbitrary
    constants given relation.
  • b) Consistent with that relation.
  • (c) Free from arbitrary constants.

4
Elimination of Arbitrary Constants.
  • The order of differential equation is equal to
    the number of arbitrary constants in the given
    relation.
  • The differential equation is consistent with the
    relation.
  • The differential equation is free from arbitrary
    constants.

5
EXAMPLE
  • Example 1.
  • y C1e-2x C2 e3x (1)
  • Y -2C1e-2x 3 C2 e3x (2)
  • Y 4C1e-2x 9 C2 e3x (3)
  • Elimination of equations 1 and 2 yields to
  • y2y 15 C2 e3x
  • The elimination of C1 from equation 1 and 2
    yields to y 2y 5 C2 e3x
  • Hence, y 2y 3(y 2y) or y c y 6y 0

NEXT
6
EXAMPLE
  • Example 2 Find the solution of xsiny x2y c
  • Solution
  • xcosy dy siny dx x2dy y2xdx 0
  • (siny 2xy)dx (xcosy x2)dy 0
  • Example 3 Find the solutionof 3x2 xy2 c
  • Solution
  • 6xdx (x2ydy y2dx)0
  • 6xdx 2xydy y2dx 0
  • (6x y2)dx 2xydy 0

TOPICS
7
FAMILIES OF CURVES
  • Obtain the differential equation of the family
    plane curves described
  • Straight lines through the origin. Answer
  • Straight lines through the fixed (h,k) h and k
    not to be eliminated. Answer
  • Straight lines with slope and x-intercept equal.
    Answer
  • Straight lines with slope and y-intercept equal.
    Answer
  • Straight lines with the algebraic sum of the
    intercept fixed as k. Answer

TOPICS
8
FAMILIES OF CURVES
  • 1. General equation
  • y mx
  • m slope
  • y m or m dy/dx
  • Substitute m,
  • Y (dy/dx)x
  • ydx xdy
  • ydx-xdy 0
  • 2. General equation
  • (y k) m ( x h )
  • dy mdx
  • m dy/dx
  • Sustitute
  • (y k) dy/dx(x-h)
  • (y k)dx (x - h)dy
  • (y k) dx ( x h) dy 0

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9
FAMILIES OF CURVES
  • 3. General Equation
  • y m(x - a)
  • m slope
  • a x-intercept
  • y m(x m)
  • dy mdx
  • mdy/dx y
  • Substitute,
  • yy (x y)
  • y xy - (y)2
  • 4. General Equation
  • y mx b
  • m slope
  • b y-interceptb m
  • y mx m
  • dy mdx
  • m dy/dx
  • Sustitute,
  • y (dy/dx)x dy/dx
  • ydx xdy dy
  • ydx (x1)dy 0

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10
FAMILIES OF CURVES
  • 5. For x intercept
  • y m(x a)
  • y m
  • y y (x a)
  • y xy ay
  • a (xy y)/y
  • For y- intercept
  • y mx b
  • y m
  • Y yx b
  • b y xy
  • But, k a b
  • K (xy y)/y (y xy)
  • Multiply by y,
  • ky xy y y (y xy)
  • ky (1 y)(xy y)
  • ky (1 y)(xy y) 0

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11
EQUATIONS OF ORDER 1
  • General Equation
  • M(x,y)dx N(x,y)dy 0
  • It can be solved by
  • Separation of Variables
  • Homogenous Equations
  • Linear Coefficients of two variables

TOPICS
12
Separation of variables
  • Solve the following
  • 1. dr/dt - 4rt when t 0, r ro Answer
  • 2. 2xyy 1 y2 when x 2 , y 3 Answer
  • 3. xyy 1 y2 when x 2, y 3 Answer
  • 4. 2ydx 3xdy when x 2, y 1 Answer
  • 5. 2ydx 3xdy when x -2, y 1 Answer

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