IBHL - Calculus - Santowski

Lesson 58 – Homogeneous Differential Equations

04/22/231 Calculus - Santowski

Lesson Objectives

Review the previous two types of FODEs that we already know how to solve

Introduce homogeneous DEs and solve using substitution

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(A) ReviewWe have seen two simple types of first order

Differential Equations so far in this course and have seen simple methods for solving them algebraically:

(1) Simple DEs in the form of wherein we use a simple antiderivative (or integral) to solve the DE

(2) DEs in the form of wherein we separate the variables in order to

solve the DE

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dydx

f (x)

dydx

f (x) g(y) OR dydx

f (x)g(y)

(A) Review - PracticeDetermine the general solution of the

following DEs

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(a) dydx

2

1 x

(b) dydx

6x 3x2 x

(c) dydx

sin2 x

(d) dydx

x sinx

ey

(e) x 1 - y2 dxdy

1

(A) Review – Introducing Slope Fields

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€

dydx

=2

1−x

€

dydx

=6x+3x2 +x

(A) Review – Introducing Slope Fields

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€

dydx

=sin2(x)

€

dydx

=xsinx

ey

Homogeneous Functions

A function f (x, y) in x and y is called a homogenous function, if the degrees of each term are equal.

Examples:

2 2g x, y = x - xy +y is a homogeneous function of degree 2

3 2 2f x, y = x +3x y +2y x is a homogeneous function of degree 3

Homogeneous Differential Equations

ƒ x, ydy=

dx g x, y

where f (x, y) and g(x, y) is a homogeneous functions of the same degree in x and y, then it is called homogeneous differential equation.

is a homogeneous differential equation as

3 2y +3xy and 3x both are homogeneous functions of degree 3.

Example:3 2

3

dy y +3xy=

dx x

(B) Homogeneous DEs A FODE in the form of is

homogeneous if it does not depend on x

and y separately, but only the ratio of y/x.

Homogeneous DEs are written in the form

ALGEBRAIC STRATEGY using a

substitution, these DEs can be turned into

separable DEs our substitution will be

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dydx

f (x,y)

dydx

fyx

v yx

OR y vx

(C) Example #1Let’s work with the

DE

But first, let’s get a visual/graphic perspective from this SLOPE FIELD diagram

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dydx

x y

x

(C) Example #1Let’s work with the DE

We will rearrange it (if possible) to a form of y/x

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dydx

x y

x

dydx

x y

xdydx

xx

yx

dydx

1yx

(C) Example #1 Now make the substitution wherein y = vx

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dydx

x y

x1

yx

let y vx

d vx dx

1vxx

ddx

vx 1v use product rule

vdxdx

xdvdx

1v rearrange

vxdvdx

1v

xdvdx

1

dvdx

1x

simple integral or separate

(C) Example #1 Now we simply integrate and simplify …..

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dvdx

1x

simple integral or separate

v ln x C but recall what v equals?

yx

ln x C replace C with lnC

yx

lnCx

y x lnCx

(C) Example #1 – Graphic Solns

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dydx

x y

x

(D) Example #2Let’s work with the

DE

But first, let’s get a visual/graphic perspective from this SLOPE FIELD diagram

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xyyx

dxdy 22

(D) Example #2Let’s work with the DE

We will rearrange it (if possible) to a form of y/x

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xyyx

dxdy 22

xy

xyx

yyx

dxdy

xyy

xyx

dxdy

xyyx

dxdy

1

22

22

(D) Example #2 Now make the substitution wherein y = vx

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dxx

vdv

vdxdv

x

vvdx

dvxv

vvdx

dvx

dxdx

v

vv

vxdxd

xvx

xvxdx

vxd

vxyxy

xydx

dy

1

separate& rearrange 1

1

1

1

rule product use now

1

let now 1

(D) Example #2 Now we simply integrate and simplify …..

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Cxxy

Cxxy

Cxv

Cxv

xvdv

ln2

ln21

equals? v what recall but lnln21

lnC with C replace ln21

integrate simply dx1

22

2

2

2

(D) Example #2 – Graphic Solns

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xyyx

dxdy 22

(E) Example #3Let’s work with the

DE

But first, let’s get a visual/graphic perspective from this SLOPE FIELD diagram

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2222 yxdxdy

x

(E) Example #3Let’s work with the DE

We will rearrange it (if possible) to a form of y/x

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2222 yxdxdy

x

22

2

2

2

2

2

22

222

121

21

21

22

2

2

xy

xy

dxdy

xy

xx

dxdy

xyx

dxdy

yxdxdy

x

(E) Example #3

Now make the substitution wherein v = y/x

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dx

xvdv

vvdxdv

x

vdxdv

xv

vdxdv

xv

vdxdv

xdxdx

v

vv

vxdxd

vdxvxd

xyv

xy

dxdy

112

separate 212

rearrange 122

121

121

1

rule product use now

121

let now 121

2

2

2

2

2

2

2

(E) Example #3 Now we simply integrate and simplify …..

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Cxx

xy

Cxxy

v

vCx

Cxv

ln2

ln2

1

1ln

2

Cxln lnln1

2

integrate simply dxx1

v-12dv

2

(E) Example #3 – Graphic Solns

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2222 yxdxdy

x

(F) Practice Problems

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0yex )(

2 )(

)(

)(

2 )(

2 )(

)(

xy

32

22

2

22

22

332

dyxedxg

xxyyxy

dxdy

f

yxyx

dxdy

e

xyxy

dxdy

d

yxdxdy

xyc

xyydxdy

xb

yxdxdy

xya

xy

(G) Video Resources From patrickJMT:

https://www.youtube.com/watch?v=vEtEAYi2cIA

https://www.youtube.com/watch?v=-in3FyX6rtM

https://www.youtube.com/watch?v=QOhjUwiQlG4

From Mathispower4uhttps://www.youtube.com/watch?v=V_rKXsUIils

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