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41. Solve the differential equation - xy – x = 0, given that y 0 when x = 0. dx

41. Solve the differential equation - xy – x = 0, given that y 0 when x = 0. dx

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Solve all Q41, 42 explaining detailly each step

40. The table shows corresponding values of x and y obtained experimentally. It is given that x and y are connected by a relation of the form y = ax", where a and b are constants. By drawing a suitable linear graph, estimate the values of a and b to one decimal place. X 1.7 2.3 3.2 4.3 5.9 Y 28.7 115.0 525.0 2100.0 8800.0 41. Solve the differential equation dy - xy – X = 0, given that y = 0 when x = 0. %3 dx 42. i) Sketch the curve y = 3x² – 12x and hence find the area of the region bounded by the curve and the lines y = 0, x = 0 and x =5. ii) Find the volume generated by rotating completely the area bounded by the curve x = 3(y" %3D 2 1) and the lines x = 0 and x = 6 about the x-axis. 'Find also the x-coordinate of the centroid of the solid generated. 43. The table shows corresponding values of x and y obtained experimentally. By drawing a suitable graph relating log10x and log10y, show that these values support the hypothesis that x and y are connected by a relationship of the form: y =px", where p and q are constants. Use your graph to estimate the values of p and q to one decimal place. Hence estimate the value of , ydx dx. Use the trapezium rule to obtain anóther estimate for J, ydx 9- 2 4 56.6 1280.0 3493.9 7963.3 350.7 dy Y 44. Solve the differential equation xy 1+y, given that x = 2, y = 0 45. The table shows corresponding values of x and y obtained experimentally. dx 0.50 0.25 0.17 0.13 0.10 0.083 0.07 0.38 0.25 0.19 0.15 0.13 0.11 0.09 V It is given that x and y are related by an equation of the form where a and b are constants. By drawing a suitable linear graph relating - and- estimate the value of a and b correct to y two decimal places. 46. Two quantities x and y are known to be connected by a law of the form y = a (x+2)", where a and n are constants. Using the given table of values below, plot log y, against log(x+2) and hence find, to one decimal place the approximate values ofn and a. 1 Y 1.06 0.58 0.38 0.27 0.20 47. Find the area of the finite region bounded by the curve: y=xlnx, the x-axis and the lines x 2. 48. Solve the differential equation y(1+x*)-2(1+y*) = 0, given that y = 0 and x = 0. dy dx 49. The variable x and y tabulated below were obtained in an experiment and are thought to obey a law of the form:y = a(x – 1)n. 5. 6.13 9. 16 20 25 Y 6.5 7.0 7.15 7.5 61
Transcribed Image Text:40. The table shows corresponding values of x and y obtained experimentally. It is given that x and y are connected by a relation of the form y = ax", where a and b are constants. By drawing a suitable linear graph, estimate the values of a and b to one decimal place. X 1.7 2.3 3.2 4.3 5.9 Y 28.7 115.0 525.0 2100.0 8800.0 41. Solve the differential equation dy - xy – X = 0, given that y = 0 when x = 0. %3 dx 42. i) Sketch the curve y = 3x² – 12x and hence find the area of the region bounded by the curve and the lines y = 0, x = 0 and x =5. ii) Find the volume generated by rotating completely the area bounded by the curve x = 3(y" %3D 2 1) and the lines x = 0 and x = 6 about the x-axis. 'Find also the x-coordinate of the centroid of the solid generated. 43. The table shows corresponding values of x and y obtained experimentally. By drawing a suitable graph relating log10x and log10y, show that these values support the hypothesis that x and y are connected by a relationship of the form: y =px", where p and q are constants. Use your graph to estimate the values of p and q to one decimal place. Hence estimate the value of , ydx dx. Use the trapezium rule to obtain anóther estimate for J, ydx 9- 2 4 56.6 1280.0 3493.9 7963.3 350.7 dy Y 44. Solve the differential equation xy 1+y, given that x = 2, y = 0 45. The table shows corresponding values of x and y obtained experimentally. dx 0.50 0.25 0.17 0.13 0.10 0.083 0.07 0.38 0.25 0.19 0.15 0.13 0.11 0.09 V It is given that x and y are related by an equation of the form where a and b are constants. By drawing a suitable linear graph relating - and- estimate the value of a and b correct to y two decimal places. 46. Two quantities x and y are known to be connected by a law of the form y = a (x+2)", where a and n are constants. Using the given table of values below, plot log y, against log(x+2) and hence find, to one decimal place the approximate values ofn and a. 1 Y 1.06 0.58 0.38 0.27 0.20 47. Find the area of the finite region bounded by the curve: y=xlnx, the x-axis and the lines x 2. 48. Solve the differential equation y(1+x*)-2(1+y*) = 0, given that y = 0 and x = 0. dy dx 49. The variable x and y tabulated below were obtained in an experiment and are thought to obey a law of the form:y = a(x – 1)n. 5. 6.13 9. 16 20 25 Y 6.5 7.0 7.15 7.5 61
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