By F.M. Dekking, C. Kraaikamp, H.P. Lopuhaä, L.E. Meester

ISBN-10: 1852338962

ISBN-13: 9781852338961

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Note that we use the same value, t0 , of the independent variable for each condition. The next example shows how to deal with a second-order IVP. 1 that any solution of the second-order linear equation y + y = 0 has the form y(t) = A cos t +B sin t for arbitrary constants A and B. ) If a solution of this equation represents the position of a moving object relative to some ﬁxed location, then the derivative of the solution represents the velocity of the particle at time t. If we specify, for example, the initial conditions y(0) = 1 and y (0) = 0, we are saying that we want the position of the particle when we begin our study to be 1 unit in a positive direction from the ﬁxed location and we want the velocity to be 0.

Use this technique to solve the equations in Problems 12–14. 12. y − y = 2x − 3 13. (x + 2y)y = 1; y(0) = −1 √ 14. y = 4x + 2y − 1 A homogeneous equation has the form dy/dx = f (x, y), where f (x, y) can be expressed in the form g(y/x) or g(x/y)—that is, as a function of the quotient y/x or the quotient x/y alone. For dy example, by dividing numerator and denominator by x 2 , we can write the equation dx = dy 2−(y/x)2 2x 2 −y 2 3xy in y the form dx = 3(y/x) = g x . Any such equation can be changed into a separable equation by making the substitution z = y/x (or z = x/y).

B −y 2 dy a e Integrals of the form have many applications in mathematics and science, especially in problems dealing with probability and statistics. For instance, the error function erf (x) = x −y 2 √2 e dy appears in many applied problems and can be evaluated easily by any CAS. π 0 Dealing with separable equations often requires some algebraic skills and some integration intuition, although technology can help in tough situations. The next example introduces a common algebraic problem. 6 Using Partial Fractions 2 The equation dz dt + 1 = z looks simple enough but requires some algebraic manipulation to get a neat solution.

### A Modern Introduction to Differential Equations by F.M. Dekking, C. Kraaikamp, H.P. Lopuhaä, L.E. Meester

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