Advanced Engineering Mathematics, International Student - download pdf or read online

By Peter V. O'Neil

ISBN-10: 0495082376

ISBN-13: 9780495082378

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Example text

3x2 + xy − x2 y −1 y =0 17. Let be a potential function for M + Ny = 0 in some region R of the plane. Show that for any constant c, + c is also a potential function. How does the general solution of M + Ny = 0 obtained by using differ from that obtained using + c? 5 33 Integrating Factors “Most” differential equations are not exact on any rectangle. But sometimes we can multiply the differential equation by a nonzero function x y to obtain an exact equation. Here is an example that suggests why this might be useful.

2yy = 1 x = x − 1 for x > 1 2. y + y = 0 x = Ce−x 2y + e for x > 0 2x √ 2xy 4. y = for x = ± 2 2 2−x 3. y = − x x = C −e 2x x = x C x2 − 2 5. xy = x − y 6. y + y = 1 x = x2 − 3 for x = 0 2x x = 1 + Ce−x In each of Problems 7 through 11, verify by implicit differentiation that the given equation implicitly defines a solution of the differential equation. 7. 2 Separable Equations 8. xy3 − y = C y3 + 3xy2 − 1 y = 0 9. y − 4x + e = C 8x − ye − 2y + xe 2 2 xy xy 17. y = x + y y 2 = 2 xy y =0 11. tan 20.

The function therefore enabled us to solve a nonexact equation by solving an exact one. This idea is worth pursuing, and we begin by giving a name to . 5 Let M x y and N x y be defined on a region R of the plane. Then x y is an integrating factor for M + Ny = 0 if x y = 0 for all x y in R, and M + Ny = 0 is exact on R. How do we find an integrating factor for M + Ny = 0? 13) 34 CHAPTER 1 First-Order Differential Equations in this region. This is a starting point. Depending on M and N , we may be able to determine from this equation.

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Advanced Engineering Mathematics, International Student Edition by Peter V. O'Neil


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