71. Solve the differential equation x(1- y)2 = -2y given that y = 2when e Snow uiat dx y = In ()(-.

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Chapter2: Second-order Linear Odes
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Solve Q72, 71 explaining detailly each step

62. The table below shows corresponding values of x and y which approximately satisfy a
relation of the form. y = ax"
X
2
3
4
6.
7
50
250
775
1875
390
7200
Where a and n are constants. By drawing a suitable linear graph, determine the values of a
and n, correct to one decimal place.
1+x
63. Sketch the graph of
showing clearly the asymptotes and the points where the curve meets
1-x
the coordinates axes.
64. The area bounded by the curve y = 3 + 2x – x and the line y= 3 is rotated completely about
the line y = 3. Find the volume of the solid of revolution obtained.
%3D
65. Express
in partial fractions. Hence, or otherwise, solve the differential equation
(1+x)(1+3x)
dy
2(y+2)
given that y = -1 when x = 0.
dx
(1+x)(1+3x)'
66. The table below shows values of a continuous variable y corresponding to given values of x.
3
4
6.
X
13.6
27.2
54.4
108.8
217.6
Use the trapezium rule to find an estimate for , ydx(
67. The expression y= ax²+ bx, is an approximation to a relation connecting two variables x and
y, where a and b are constants. By using the values given in the table below, draw a suitable
straight line graph and use it to estimate the values of a and b.
2
3
4
6.
1
126
162
172
175
144
74
dy
= xy, given that y = 1, when x= 0
68. Solve the differential equation: (1 + x2)
dx
69. (In this question, you are advised to work throughout with 2 decimal places.)
The table below shows corresponding values ofx and y obtained in a certain experiment.
2.39
4.16
6.31
8.70
12.01
1.59
6.91
16.59
31.70
52.48
87.10
3.63
The relation connecting x and y is: y = x" where'2 and n are constants.
By drawing a suitable linear graph relating log10 xand log10 y, calculate the values of 1 and
n.
dy
X =
-
70. Obtain in the form: y = f(x), a particular solution of the differential equation:
dx
2xy, given that y = 0 when x = 0.
71. Solve the differential equation x(1 – y) = -2y given that y = 2when x = e. Hence, show that
dx
dy
y = In (2)(-
dy
72. Solve the differential equation x
dx
y(2x² + 1)(.
Transcribed Image Text:62. The table below shows corresponding values of x and y which approximately satisfy a relation of the form. y = ax" X 2 3 4 6. 7 50 250 775 1875 390 7200 Where a and n are constants. By drawing a suitable linear graph, determine the values of a and n, correct to one decimal place. 1+x 63. Sketch the graph of showing clearly the asymptotes and the points where the curve meets 1-x the coordinates axes. 64. The area bounded by the curve y = 3 + 2x – x and the line y= 3 is rotated completely about the line y = 3. Find the volume of the solid of revolution obtained. %3D 65. Express in partial fractions. Hence, or otherwise, solve the differential equation (1+x)(1+3x) dy 2(y+2) given that y = -1 when x = 0. dx (1+x)(1+3x)' 66. The table below shows values of a continuous variable y corresponding to given values of x. 3 4 6. X 13.6 27.2 54.4 108.8 217.6 Use the trapezium rule to find an estimate for , ydx( 67. The expression y= ax²+ bx, is an approximation to a relation connecting two variables x and y, where a and b are constants. By using the values given in the table below, draw a suitable straight line graph and use it to estimate the values of a and b. 2 3 4 6. 1 126 162 172 175 144 74 dy = xy, given that y = 1, when x= 0 68. Solve the differential equation: (1 + x2) dx 69. (In this question, you are advised to work throughout with 2 decimal places.) The table below shows corresponding values ofx and y obtained in a certain experiment. 2.39 4.16 6.31 8.70 12.01 1.59 6.91 16.59 31.70 52.48 87.10 3.63 The relation connecting x and y is: y = x" where'2 and n are constants. By drawing a suitable linear graph relating log10 xand log10 y, calculate the values of 1 and n. dy X = - 70. Obtain in the form: y = f(x), a particular solution of the differential equation: dx 2xy, given that y = 0 when x = 0. 71. Solve the differential equation x(1 – y) = -2y given that y = 2when x = e. Hence, show that dx dy y = In (2)(- dy 72. Solve the differential equation x dx y(2x² + 1)(.
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