By F.M. Dekking, C. Kraaikamp, H.P. Lopuhaä, L.E. Meester
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Additional info for A Modern Introduction to Differential Equations
5/3600) of an hour. 291667 . . mile ≈ 154 feet. We have shown that in going from 0 to 60, the 2005 Ferrari F430 will travel approximately 154 feet. ■ General Solutions If every solution of an nth-order differential equation F x, y, y , y , y , . . , y (n−1) , y (n) = 0 on an interval (a, b) can be obtained from an n-parameter family by choosing appropriate values for the n constants, we say that the family is the general solution of the differential equation. In this case, we will need n initial conditions or n boundary conditions (or a combination of n conditions) to determine the constants.
In Chapters 2 and 3, we’ll discuss the question of existence and uniqueness for IVPs: Does the equation or system have a solution that satisﬁes the initial conditions? If so, is there only one solution? If every solution of an nth-order ODE on an interval can be obtained from an n-parameter family by choosing appropriate values for the n constants, then we say that the family is the general solution of the differential equation. In this case we need n initial conditions or n boundary conditions to determine the constants.
2. The relationship between the velocity v of a riﬂe bullet and the distance L traveled by it in the aLn , where v = dL and n < 1. barrel of the gun is established in ballistics by the equation v = b+L n dt Find the relationship between the time t during which the bullet moves in the barrel and the distance L covered. 3. In trying to determine the shape of a ﬂexible nonstretching cable suspended between two points A and B of equal height, we can analyze the forces acting on the cable and get the differential equation d2 y =k 1+ dx 2 1/2 dy 2 , dx where k > 0 is a constant.
A Modern Introduction to Differential Equations by F.M. Dekking, C. Kraaikamp, H.P. Lopuhaä, L.E. Meester