56. Given that x+y - xy and that y dx dy when x 1 show that xy = e*

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Chapter1: Functions And Models
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Solve Q56, 57, 58 explaining detailly each step

54. The table below shows corresponding values of p and q, two variables thought to obey a law
of the form q = ae*P, where a and k are real constants
%3D
10
15
20
8.12
8.62
9.30
10.02
10.80
Show that such a relation is possible by transforming the formula into a suitable linear form.
By drawing the line, estimate values for a andk correct to two decimal places.
55. Solve the differential equation: (x - 1)+ 2y = 0 given that y = 3 when x =2. expressing
dy
dx
your answer in form y f(x).
dy
56. Given that x- +y - xy and that y
when x 1 show that xy = e*-2.
dx
d²y
dy
57. Given that e*= cos4x, show that:
dx2
-3x,
-6- + 25y = 0.
dx
dy
+ 1 = y and that y, = 2 when x
1
58. Given that: 2x
dx
1+x
show that y
3
1-x
59. Solve in the form y f(x), the differential equations: (1-x)!
dy
2xy 0 given that
dx
y = 1 when x = 0
60. The values of the two variables x and y shown below, as obtained from a particular
a
experiment, obey a law of the form y ==+b where a and b are constants.
By piotting a suitable straight line graph, determine the values of a and b, giving your answer to
one decimal piace. Obtain also an estimate for the values of x when y =4.5
1
1.5
3
4
6.
8.
10
5.2
4.8
4.4
4.2
4
3.8
3.84
Transcribed Image Text:54. The table below shows corresponding values of p and q, two variables thought to obey a law of the form q = ae*P, where a and k are real constants %3D 10 15 20 8.12 8.62 9.30 10.02 10.80 Show that such a relation is possible by transforming the formula into a suitable linear form. By drawing the line, estimate values for a andk correct to two decimal places. 55. Solve the differential equation: (x - 1)+ 2y = 0 given that y = 3 when x =2. expressing dy dx your answer in form y f(x). dy 56. Given that x- +y - xy and that y when x 1 show that xy = e*-2. dx d²y dy 57. Given that e*= cos4x, show that: dx2 -3x, -6- + 25y = 0. dx dy + 1 = y and that y, = 2 when x 1 58. Given that: 2x dx 1+x show that y 3 1-x 59. Solve in the form y f(x), the differential equations: (1-x)! dy 2xy 0 given that dx y = 1 when x = 0 60. The values of the two variables x and y shown below, as obtained from a particular a experiment, obey a law of the form y ==+b where a and b are constants. By piotting a suitable straight line graph, determine the values of a and b, giving your answer to one decimal piace. Obtain also an estimate for the values of x when y =4.5 1 1.5 3 4 6. 8. 10 5.2 4.8 4.4 4.2 4 3.8 3.84
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