Differential Equations

Download PDF by Florian Cajori: An introduction to the theory of equations

By Florian Cajori

ISBN-10: 0486621847

ISBN-13: 9780486621845

ISBN-10: 1418165557

ISBN-13: 9781418165550

Initially released in 1904. This quantity from the Cornell college Library's print collections was once scanned on an APT BookScan and switched over to JPG 2000 structure via Kirtas applied sciences. All titles scanned hide to hide and pages might comprise marks notations and different marginalia found in the unique quantity.

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Yn(x)} is linearly independent on a ≤ x ≤ b. 2. The nth-order linear homogeneous differential equation L(y) = 0 always has n linearly independent solutions. ,cn denote arbitrary constants. 3. If the Wronskian of a set of n functions defined on the interval a ≤ x ≤ b is nonzero for at least one point in this interval, then the set of functions is linearly independent there. If the Wronskian is identically zero on this interval and if each of the functions is a solution to the same linear differential equation, then the set of functions is linearly dependent.

Click Here for Terms of Use. 48 DIFFERENTIAL EQUATIONS Figure 7-1 Hooke’s law: The restoring force F of a spring is equal and opposite to the forces applied to the spring and is proportional to the extension (contraction) l of the spring as a result of the applied force; that is, F = −kl, where k denotes the constant of proportionality, generally called the spring constant. 1. A steel ball weighing 128 lb is suspended from a spring, whereupon the spring is stretched 2 ft from its natural length.

1 and simplifying. 2 can be easily obtained by replacing y ′′, y ′, and y by l2, l1, and, l0 = 1, respectively. 1. 1 The characteristic equation of y ′′ + 3 y ′ − 4 y = 0 is l2 + 3l − 4 = 0; the characteristic equation of y ′′ − 2 y ′ + y = 0 is l2 − 2l + 1 = 0. Characteristic equations for differential equations having dependent variables other than y are obtained analogously, by replacing the jth derivative of the dependent variable by l j ( j = 0,1,2). , n). , n). 2. The characteristic equation of y ( 4 ) − 3 y ′′′ + 2 y ′′ − y = 0 is l4 − 3l3 + 2l2 − 1 = 0.

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An introduction to the theory of equations by Florian Cajori

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