Solutions by Substitutions, p' 7578 2'5

1
Solutions by Substitutions, p. 75-78 (2.5)

• OBJECTIVES
• Define homogeneous DE
• Use substitutions to solve DE
• Solve Bernoullis Equation

2

• Consider the separable DE y dx x dy 0
• Let y vx or dy x dv v dx
• Substitution vx dx x (x dv v dx) 0
• or vx dx x2 dv vx dx 0
• x2 dv -2vx dx
• Separating variables
• Integrating ln v 2ln x k , k constant
• ln v ln x2 k

3

• homogeneous function a function f with the
  property for some real number , p. 75.
• 1st order homogeneous DE an equation of the
  form M (x, y) dx N(x, y) dy 0 with M, and N
  are homogeneous fo the same degree, or
• and
• p. 75.
• Homogeneous equations can be solved by
  substitution of y ux or x vy, p. 75.

4

• Consider (xy y2 x2) dx x2 dy 0.
• Note M (x, y) xy y2 x2 , N(x, y) x2
• (tx)(ty) (ty) 2 (tx) 2,
• t2xy t2 y2 t2x2,
• t2 (xy y2 x2),
• t2 M(x, y),

(tx)2 t2x2 t2x2
t2 N(x, y)
5

• Consider (xy y2 x2) dx x2 dy 0.
• Let y ux, then dy x du udx.
• Substitution
• x (ux) (ux) 2 x2 dx x2 (x du udx) 0
• (x2 u u2 x2 x2)dx x3 du x2 udx 0
• (u2 x2 x2)dx x3 du 0
• (u2 1) x2dx x3 du

6

• (u2 1) x2dx x3 du
• ln x arctan u k, k constant
• u tan(ln x k) or y x tan(ln x
  k)

7
Method of solution, p. 75

• a) Substitute either y ux or Nx vy .
• b) Rewrite equation using new variables.
• c) Solve transformed equation.
• d) Express solution in terms of original
  variables.

8

• Consider
• Let u x y, then or
• Substitution

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• Integrating
• (u 2)2 e2(x k) 1
• (x y 1)2 Ce2x 1, C ek

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• Bernoulli Equation
• Consider
• Let u y2 then or
• Substitution

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• Multiply by
• integrating factor e10x,
• Separating variables
• Integrating by parts
• k constant

12

• C k C1 , C1 constant
• Multiply by e10x,
• Implicit solution

13
Homework

• p. 10 21-24
• p. 17 15-24
• p. 29 9-12
• p. 78 1-30 alternate odd
• Read p. 92-98 (3.1)
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