. Differential and integral calculus, an introductory course for colleges and engineering schools. and each curve of whichsatisfies the requirements of the problem. C is termed the parameter ofthis system of curves. By giving a suitable value to C, we may select a curve of the system tosatisfy some other condition, as, for example, that it should pass throughthe point (2, 1). Substituting 2 and 1 for x and y in (a), we have 1 = 2 a + C, whence C = 1 - 2 y=±ax2 + l-2a is the particular curve of the system that passes through (2, 1). From the foregoing solution it is evident that every eq
. Differential and integral calculus, an introductory course for colleges and engineering schools. and each curve of whichsatisfies the requirements of the problem. C is termed the parameter ofthis system of curves. By giving a suitable value to C, we may select a curve of the system tosatisfy some other condition, as, for example, that it should pass throughthe point (2, 1). Substituting 2 and 1 for x and y in (a), we have 1 = 2 a + C, whence C = 1 - 2 y=±ax2 + l-2a is the particular curve of the system that passes through (2, 1). From the foregoing solution it is evident that every equationin x and y that arises from an integration contains the constant ofintegration as a parameter, and consequently represents a systemof curves. 201 202 INTEGRAL CALCULUS §145 Problem 2. To determine the system of curves whose subtangent The equation of the tangent to a curve is y — 2/1 = Dyi(x - Xi).The subtangent is the line AB of therpf figure. If x be the abscissa of A, AB = Xi — x. When y = 0 in the equationof the tangent, x = x; therefore-2/i = Ztyi(z - xi),. Xi — x = ~- = andNow by the condition of the problem where a is a known constant. Therefore, omitting subscripts, aDy = y, ady = y dx, a dy y dx. Hence and ./?-/*+*, 2/ a log 7/ = x + C, C being the constant of integration. This is the equation of the systemsought. It may be transformed as follows: log y = x + C x + C C x = e° ea, or, writing k for the constant e°, X y = keai and is now seen to be the equation of a system of exponential parameter is k. This is the complete solution of the problem. Asin the preceding problem, a curve of the system may be found to fulfillsome further condition. For example, if the curve is to pass through thepoint (0, 1), we have 1 = ke° = k, and hence y = ea is that curve of the system which passes through the point (0, 1). §145 APPLICATIONS OF INTEGRATION IN GEOMETRY 203 Problem 3. To determine the system of curves in whic
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