Elementary differential equation
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Transcript of Elementary differential equation
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
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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