Explanation Given: 4 y y ′ − x = 0 , 4 y 2 − x 2 = C ; C = 0 , C = ± 1 , C = ± 4 . Graph : It is given that the general equation of the differential equation is 4 y 2 − x 2 = C .......... (1) Putting C = 0 , in equation (1), we get 4 y 2 − x 2 = 0 —— (2) Now, rearranging equation (2) gives us, y = ± 1 2 x We start by creating a table showing the values of y at different values of x . The table shown is asmall sample. x 0 1 2 3 4 -2 y = 1 2 x 0 1 2 1 3 2 2 1 y = − 1 2 x 0 − 1 2 − 1 − 3 2 − 2 − 1 Now, using a graphing utility tool, the plot of 4 y 2 − x 2 = 0 is given by The graph of y = 1 2 x is a straight line that passes through the origin and has a positive slope. The graph of y = − 1 2 x is a straight line that passes through the origin and has a negative slope. Putting C = ± 1 , in equation (1), we get 4 y 2 − x 2 = ± 1 . We start by creating a table showing the values of y at different values of x . The table shown is a small sample. x 0 1 2 3 4 -2 y = 1 2 x 2 + 1 1 2 1 2 5 2 10 2 17 2 10 2 y = − 1 2 x 2 + 1 − 1 2 − 1 2 − 5 2 − 10 2 − 17 2 − 10 2 y = 1 2 x 2 − 1 0 3 2 2 15 2 3 2 y = − 1 2 x 2 − 1 0 − 3 2 − 2 − 15 2 − 3 2 From the table, the corresponding co-ordinate points are plotted on a graph paper and then the points are connected to obtain the particular solution curve, which represents a hyperbola

11th Edition

ISBN: 9781337275361

Author: Ron Larson, Bruce H. Edwards

Publisher: Cengage Learning

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Differential Equations

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Chapter 6.1, Problem 35E

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To Graph: The particular solution of 4y2−x2=C for the given values of C.

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Chapter 6.1, Problem 34E

Chapter 6.1, Problem 36E

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To Graph: The particular solution of 4 y 2 − x 2 = C for the given values of C .

Solution Summary: The author explains how the graph of 4y2-x2=C is given using a graphing utility tool.

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To Graph: The particular solution of 4y2−x2=C for the given values of C.

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