Elimination of Arbitrary

Constants

Problem 01

Solution 01

Divide by 3x

answer

Problem 02

Solution 02

answer

Problem 03

Solution 03

→ equation (1)

Divide by dx

Substitute c to equation (1)

Multiply by dx

answer

Another Solution

okay

Problem 04

Solution 04

→ equation (1)

Substitute c to equation (1)

answer

Another Solution

Divide by y2

Multiply by y3

okay

Separation of Variables | Equations of Order

One

Problem 01

, when ,

Solution 01

HideClick here to show or hide the solution

when ,

then,

answer

Problem 02

, when , .

Solution 2

when ,

then,

answer

Problem 03

, when , .

Solution 03

when ,

then,

answer

Problem 04

, when , .

Solution 04

Therefore,

when x = 2, y = 1

Thus,

answer

Problem 05

, when , .

Solution 05

From Solution 04,

when x = -2, y = 1

Thus,

answer

Problem 06

, when , .

Solution 06

From Solution 04,

when x = 2, y = -1

Thus,

answer

Problem 07

, when , .

Solution 07

when x = 0, y = 0

thus,

answer

Problem 08

, when

, .

Solution 08

For

Let

,

,

Then,

when x → ∞, y → ½

Thus,

answer

Problem 09

, when

, .

Solution 09

when θ = 0, r = a

Thus,

answer

Problem 10

, when , .

Solution 10

when x = xo, v = vo

thus,

answer

Problem 11

Solution 11

answer

Problem 12

Solution 12

answer

Problem 13

Solution 13

answer

Problem 14

Solution 14

answer

Problem 15

Solution 15

answer

Problem 16

Solution 16

answer

Problem 17

Solution 17

answer

Problem 18

Solution 18

answer

Problem 19

Solution 19

answer

Problem 20

Solution 20

answer

Problem 21

Solution 21

By long division

Thus,

ans

wer

Problem 22

Solution 22

By long division

Thus,

answer

Problem 23

Solution 23

answer

Homogeneous Functions | Equations of Order

One

Problem 01

Solution 01

Let

Substitute,

Divide by x2,

From

Thus,

answer

Problem 02

Solution 02

Let

Substitute,

From

answer

Problem 03

Solution 03

Let

From

Thus,

answer

Problem 04

Solution 04

Let

From

Thus,

answer

Exact Equations | Equations of Order One

Problem 01

Solution 01

Test for exactness

;

;

; thus, exact!

Step 1: Let

Step 2: Integrate partially with respect to x,

holding y as constant

→ Equation (1)

Step 3: Differentiate Equation (1) partially with

respect to y, holding x as constant

Step 4: Equate the result of Step 3 to N and

collect similar terms. Let

Step 5: Integrate partially the result in Step 4

with respect to y, holding x as constant

Step 6: Substitute f(y) to Equation (1)

Equate F to ½c

answer

Problem 02

Solution 02

Test for exactness

Exact!

Let

Integrate partially in x, holding y as constant

→ Equation (1)

Differentiate partially in y, holding x as constant

Let

Integrate partially in y, holding x as constant

Substitute f(y) to Equation (1)

Equate F to c

answer

Problem 03

Solution 03

Test for exactness

Exact!

Let

Integrate partially in x, holding y as constant

→ Equation (1)

Differentiate partially in y, holding x as constant

Let

Integrate partially in y, holding x as constant

Substitute f(y) to Equation (1)

Equate F to c

answer

Problem 04

Solution 04

Test for exactness

Exact!

Let

Integrate partially in x, holding y as constant

→ Equation

(1)

Differentiate partially in y, holding x as constant

Let

Integrate partially in y, holding x as constant

Substitute f(y) to Equation (1)

Equate F to c

answer

Linear Equations of Order One

Problem 01

Solution 01

→ linear in y

Hence,

Integrating factor:

Thus,

Multiply by 2x3

answer

Problem 02

Solution 02

→ linear in

y

Hence,

Integrating factor:

Thus,

Mulitply by (x + 2)-4

answer

Problem 03

Solution 03

→ linear in y

Hence,

Integrating factor:

Thus,

Using integration by parts

,

,

Multiply by 4e-2x

answer

Problem 04

Solution 04

→ linear in x

Hence,

Integrating factor:

Thus,

Using integration by parts

,

,

Multiply 20(y + 1)-4

answer

Integrating Factors Found by Inspection

Problem 01

Solution 01

Divide by y2

Multiply by y

answer

Problem 02

Solution 02

Divide by y3

answer

Problem 03

Solution 03

Divide by x both sides

answer

Problem 04

Solution 04

Multiply by s2t2

answer

Problem 05

Problem 05

answer

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Problem 06

Solution 06

answer

Problem 07

Solution 07 - Another Solution for Problem 06

Divide by xy(y2 + 1)

Resolve into partial fraction

Set y = 0, A = -1

Equate coefficients of y2

1 = A + B

1 = -1 + B

B = 2

Equate coefficients of y

0 = 0 + C

C = 0

Hence,

Thus,

answer - okay

Problem 11

Solution 11

answer

The Determination of Integrating Factor

Problem 01

Solution 01

→ a function of

x alone

Integrating factor

Thus,

answer

Problem 02

Solution 02

→

a function of x alone

Integrating factor

Thus,

answer

Problem 03

Solution 03

→

neither a function of x alone nor y alone

→ a function of y alone

Integrating factor

Thus,

answer

Problem 04

Solution 04

→

neither a function of x alone nor y alone

→

a function y alone

Integrating factor

Thus,

answer

Substitution Suggested by the Equation |

Bernoulli's Equation

Problem 01

Solution 01

Let

Thus,

→ vari

ables separable

Divide by ½(5z + 11)

ans

Problem 02

Solution 02

Let

Hence,

→ homogeneo

us equation

Let

Divide by vx3(3 + v)

Consider

Set v = 0, A = 2/3

Set v = -3, B = -2/3

Thus,

From

But

answer

Problem 03

Solution 03

Let

But

answer

Problem 04

Solution 04

→ Bernoulli's

equation

From which

Integrating factor,

Thus,

answer

Problem 05

Solution 05

Let

answer

Elementary Applications

Newton's Law of Cooling

Problem 01

A thermometer which has been at the reading

of 70°F inside a house is placed outside where

the air temperature is 10°F. Three minutes later

it is found that the thermometer reading is

25°F. Find the thermometer reading after 6

minutes.

Solution 01

According to Newton’s Law of cooling, the

time rate of change of temperature is

proportional to the temperature difference.

When t = 0, T = 70°F

Hence,

When t = 3 min, T = 25°F

Thus,

After 6 minutes, t = 6

answer

Simple Chemical Conversion

Problem 01

Radium decomposes at a rate proportional to

the quantity of radium present. Suppose it is

found that in 25 years approximately 1.1% of

certain quantity of radium has decomposed.

Determine how long (in years) it will take for

one-half of the original amount of radium to

decompose.

Solution 01

When t = 25 yrs., x = (100% - 1.1%)xo = 0.989xo

Thus,

When x = 0.5xo

answer

Problem 02

A certain radioactive substance has a half-life of

38 hour. Find how long it takes for 90% of the

radioactivity to be dissipated.

Solution 02

When t = 38 hr, x = 0.5xo

Hence,

When 90% are dissipated, x = 0.1xo

answer