Aim: Separation of variables: Divorce – Calculus style!

Aim: Separation of Variables Course: Calculus

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Aim: Separation of variables: Divorce – Calculus style!

1

Use log differentiation to find : for

2x

dy

dx

y x


Separation of Variables

When all x terms are collected with dx and all y terms are collected with dy on opposite sides of a differential equation.

original separated

2 3 0dy

x ydx

23 ydy x dx

sin ' cosxy x cotdy x dx

'2

1y

xy

e

1 2

1y dy dxe x


Model Problem

4If and (0) 5 find an equation

for in terms of

dy xy

dx y

y x

4y dy x dx

4y dy x dx 2 2

2 21 22 2

2 2

y yC x C x C

2

25 252 0

2 2C C

22 25

22 2

yx

Separate, integrate, and solve for constant


Model Problem

2If 3 and (0) 2 find an equation

for in terms of

dyx y y

dxy x

23dy

x dxy

Separate, integrate, and solve for constant

23dy

x dxy

3ln y x C 3x Cy e 1CC e

3xy Ce


Model Problem

2If 3 and (0) 2 find an equation for in terms of

dyx y y y x

dx

3xy Ce302 2Ce C

3

2 xy e


Model Problem – General Solution

Find a general solution of 2 4dy

x xydx

2 4x dy xydx

2 4

dy x dx

y x

2 4

dy x dx

y x

21

1ln ln 4

2y x C 2

1ln 4x C

1 12 24 4C Cy e x y e x

2 4y C x

differential form

separate variables

integrate both sides

general solution

1CC e


Model Problem – Particular Solution

Find the equation of a curve that passes through (1, 3) and has a slope of y/x2 at the

point (x, y).

2

dy y

dx x y(1) = 3

2

dy dx

y x 2

dy dx

y x 1

1ln y C

x

11/ 1/x C xy e Ce 13 3Ce C e

1/ 1 /3 3 , 0x x xy e e e x

separate integrate antiderivative

solve for y

1CC e

Particular Solution

General Solution


Model Problem

If the acceleration of a particle is given by a(t) = -32 ft/sec2, and the velocity of the particle is 64 ft/sec and the height of the particle is 32 ft at time t = 0, find: a) the equation of the particle’s velocity at time t; b) the equation for the particle’s height, h at time t; and c) the maximum height of the particle

dv

dt( ) 32a t

32dv dt 32dv dt


Model Problem

If the acceleration of a particle is given by a(t) = -32 ft/sec2, and the velocity of the particle is 64 ft/sec and the height of the particle is 32 ft at time t = 0, find:

a) the equation of the particle’s velocity at time t; 32dv dt

32v t C

64 32 0 64C C

32 64v t


Model Problem

If the acceleration of a particle is given by a(t) = -32 ft/sec2, and the velocity of the particle is 64 ft/sec and the height of the particle is 32 ft at time t = 0, find:

b) the equation for the particle’s height, h at time t; dh

dt32 64v t

32 64dh t dt

32 64dh t dt 216 64h t t C

232 16 0 64 0 C 32C

216 64 32h t t


Model Problem

If the acceleration of a particle is given by a(t) = -32 ft/sec2, and the velocity of the particle is 64 ft/sec and the height of the particle is 32 ft at time t = 0, find: c) the maximum height of the particle

032 64v t 0dh

dt

216 64 32h t t

2t

216 2 64 2 32h 96h


Model Problem

A city had a population of 10,000 in 1980 and 13,000 in 1990. Assuming an exponential growth rate, estimate the city’s population in 2000.

dy

dtky

Separate dy

k dty

dyk dt

y ln y kt C

kt Cy e

1CC ekty Ce

integrate


Model Problem

A city had a population of 10,000 in 1980 and 13,000 in 1990. Assuming an exponential growth rate, estimate the city’s population in 2000.

kty Ce

1980: t = 0 1990: t = 10 2000: t = 20

solve

1ln1.3 0.262

10k

0.026210000 ty e

010000 kCe 1013000 kCe

16,900 - population in 2000y

1013000 10000 ke



Find a particular solution given y(0) = 1 of

2 2 1 0xxydx e y dy

2

2 2 1 0x

xexydx e y dy

y

22 1

0xy dy

e x dxy

22 1

xy dy

e x dxy

22 1

xy

dy e x dxy



Find a particular solution given y(0) = 1 of

2 2 1 0xxydx e y dy

22 1

xy

dy e x dxy

21 xy dy xe dxy

22 1

ln2 2

xyy e C

22

01 1 1 1ln1 0

2 2 2 2e C C

C = 12 2

22 21

ln 1 ln 22 2

x xyy e y y e

Aim: Separation of variables: Divorce – Calculus style!

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