1. Elimination of Arbitrary
Constants
→ equation (1)
Problem 01
Divide by dx
Solution 01
Substitute c to equation (1)
Divide by 3x
Multiply by dx
answer
Problem 02
answer
Another Solution
Solution 02
answer
okay
Problem 03
Problem 04
Solution 03
2. Solution 04
→ equation (1)
Multiply by y3
Substitute c to equation (1)
okay
Separation of Variables | Equations of Order
One
Problem 01
, when
,
Solution 01
answer
HideClick here to show or hide the solution
Another Solution
Divide by y2
when
,
4. answer
Problem 04
, when
,
answer
.
Problem 05
, when
Solution 04
,
.
Solution 05
From Solution 04,
when x = -2, y = 1
Thus,
Therefore,
answer
when x = 2, y = 1
Problem 06
, when
Thus,
Solution 06
,
.
5. From Solution 04,
when x = 0, y = 0
when x = 2, y = -1
thus,
Thus,
answer
Problem 07
, when
,
.
Solution 07
answer
Problem 08
, when
,
.
Solution 08
11. Problem 23
Solution 23
answer
Homogeneous Functions | Equations of Order
One
Problem 01
From
Solution 01
Thus,
Let
answer
Substitute,
Problem 02
Divide by x2,
Solution 02
14. Test for exactness
Step 5: Integrate partially the result in Step 4
with respect to y, holding x as constant
;
;
; thus, exact!
Step 1: Let
Step 6: Substitute f(y) to Equation (1)
Equate F to ½c
answer
Step 2: Integrate partially with respect to x,
holding y as constant
Problem 02
Solution 02
→ Equation (1)
Step 3: Differentiate Equation (1) partially with
respect to y, holding x as constant
Test for exactness
Step 4: Equate the result of Step 3 to N and
collect similar terms. Let
Exact!
Let
15. Solution 03
Integrate partially in x, holding y as constant
→ Equation (1)
Test for exactness
Differentiate partially in y, holding x as constant
Exact!
Let
Let
Integrate partially in y, holding x as constant
Integrate partially in x, holding y as constant
→ Equation (1)
Substitute f(y) to Equation (1)
Differentiate partially in y, holding x as constant
Equate F to c
answer
Problem 03
Let
16. Integrate partially in y, holding x as constant
Integrate partially in x, holding y as constant
→ Equation
Substitute f(y) to Equation (1)
(1)
Differentiate partially in y, holding x as constant
Equate F to c
answer
Problem 04
Let
Solution 04
Integrate partially in y, holding x as constant
Test for exactness
Substitute f(y) to Equation (1)
Exact!
Equate F to c
answer
Let
Linear Equations of Order One
17. Problem 01
Solution 01
Multiply by 2x3
answer
Problem 02
Solution 02
→ linear in y
Hence,
Integrating factor:
→ linear in
y
Thus,
Hence,
19. → linear in x
Multiply 20(y + 1)-4
Hence,
answer
Integrating Factors Found by Inspection
Problem 01
Integrating factor:
Solution 01
Divide by y2
Thus,
Using integration by parts
,
Multiply by y
answer
,
Problem 02
20. Divide by x both sides
Solution 02
Divide by y3
answer
Problem 04
Solution 04
answer
Problem 03
Solution 03
21. - See more at:
Problem 06
Solution 06
Multiply by s2t2
answer
Problem 05
Problem 05
answer
Problem 07
Solution 07 - Another Solution for Problem 06
Divide by xy(y2 + 1)
answer
22. Resolve into partial fraction
Problem 11
Solution 11
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,
answer
The Determination of Integrating Factor
Problem 01
Thus,
Solution 01
answer - okay
23. answer
→ a function of
x alone
Problem 02
Solution 02
Integrating factor
Thus,
24. →
a function of x alone
Integrating factor
Thus,
→
neither a function of x alone nor y alone
→ a function of y alone
answer
Problem 03
Integrating factor
Solution 03
Thus,
25. →
neither a function of x alone nor y alone
→
a function y alone
answer
Problem 04
Integrating factor
Solution 04
Thus,
26. answer
Substitution Suggested by the Equation |
Bernoulli's Equation
Problem 01
Solution 01
Let
ans
Problem 02
Thus,
Solution 02
Let
→ vari
ables separable
Divide by ½(5z + 11)
Hence,
→ homogeneo
us equation
Let
27. Divide by vx3(3 + v)
From
Consider
Set v = 0, A = 2/3
Set v = -3, B = -2/3
But
Thus,
answer
Problem 03
29. answer
Problem 05
answer
Solution 05
Elementary Applications
Newton's Law of Cooling
Let
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
30. Thus,
Hence,
When x = 0.5xo
When t = 3 min, T = 25°F
answer
Thus,
After 6 minutes, t = 6
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
Simple Chemical Conversion
When t = 38 hr, x = 0.5xo
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
Hence,
When 90% are dissipated, x = 0.1xo
answer
When t = 25 yrs., x = (100% - 1.1%)xo = 0.989xo