Chapter Notes (2)

# See example 22 dx 1 by computing 2 verify that for any

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Unformatted text preview: y = xe x + Ce x is a solution of dx dy the differential equation = e x + y . (See Example 2.2.) dx 1. By computing 2. Verify that for any constant C the expression y = x 2 − 2 x + C is a solution of the differential dy x − 1 = . equation dx y 3. a) Verify that for any constant C the expression y = x − 1 + Ce − x is a solution of the dy differential equation = x− y. dx b) Find the solution of the differential equation in 3a) that satisfies y (0) = 2 . 4. a) Verify that for any constant C the expression y = equation dy 1 + xy = . dx 1 + x Ce x − 1 is a solution of the differential 1+ x b) Find the solution of the differential equation in 4a)that satisfies y (1) = 2 . 5. a) Verify that for any constant C ≥ 0 the expression y = − x + 2 x 2 + C is a solution of the dy x − y differential equation = . dx x + y b) Find the solution of the differential equation in 5a) that satisfies y (1) = 1 . 6. Which, if any, of the differential equations in exercises 1 - 5 are separable? 7. Which, if any, of the following differential equations are separable? 27 2 Differential Equations a) dy xy = dx 1 + x b) dy xy = dx y + x c) dy = (3 y + 1) 2 dx d) dy 1 + y 2 = dx 1 + x 2 Use separation of variables to solve each of the differential equations in exercises 8 - 16 below: 8. dy = xy , y (0) = 2 . dx 9. dy = 2 y 2 x , y (0) = 2 dx 10. dy = dt 11. dy t = 2 y , y (0) = 1 . dt e 12. dy = 3 y + 1 , y (1) = 2 . dt 13. y′ = y ( x + 1) , y (0) = 2 . 14. dy 2y = , y (0) = 3 . dx x + 1 15. y′ = 16. dy 1 + 2 y = dx 1 + x t , y (1) = 3 y +1 2(t + 1) , y (0) = 2 . y 17. After 5 years, which of the following accounts will have a larger principal? After 10 years? Justify your answers. a) An account earning 5% annual interest compounded continuously and starting out with \$1000. b) An account earning 4% annual interest compounded continuously and starting out with \$1100. 18. a) Suppose you want to put away a certain amount of money so that in 10 years the principal will be \$10,000. If you can invest at 6% annual interest compounded conti...
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## This document was uploaded on 04/06/2014.

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