Differential Equations

# Differential Equations - Contents 15 Differential Equations...

This preview shows pages 1–5. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Contents 15 Differential Equations 222 15.1 Simple Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 15.2 Geometric Aspects of Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 15.3 Solving Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 15.4 Exact Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 15.5 Linear Equations of First Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 15.6 Homogeneous Linear Equations with Constant Coefficients . . . . . . . . . . . . . . . . . . . . 230 CONTENTS 2 Chapter 15 Differential Equations 15.1 Simple Differential Equations Verify that the differential equation in question 1 to 16 has the given function as a solution. 1. x dy dx + y = x 2 ; f ( x ) = 1 3 x 2 + c x NOTE: y = f ( x ) and f ( x ) = dy dx Solution: The derivative of f is f ( x ) = 2 x 3- c x 2 . Substitute in for dy dx x 2 3 x- c x 2 + 1 3 x 2 + c x = x 2 . Equation is valid for all x for which f ( x ) is defined, namely x 6 = 0. 2. dy dx = x 2 ; f ( x ) = 1 3 x 3 3. dy dx = 3 x + 5; f ( x ) = 3 2 x 2 + 5 x- 2 4. ds dt = 1 √ t- 1 ; f ( t ) = 2( √ t- 1 + 1) 5. x 3 y = x 4- √ 3; g ( x ) = 1 2 x 2 + √ 3 2 x 2 6. d 2 y dx 2 = x,y = 1 6 ( x 3 + 2 x- 2) 7. d 2 s dt 2- 2 ds dt + 10 = 0; s = 5 t 8. xy = 2 y ; y = Cx 2 9. y 000- y 00 = 0; y = C 1 + C 2 x 10. y dy dx = x 2 , y = q 2 3 x 3 + 4 11. u x du dx = 1; u = √ x 2 + k 12. dy dx = y 2 x 2 ; y = x 1- ax 13. x ( y ) 2- yy + 1 = 0; y = C + x C 223 15.1 Simple Differential Equations 14. dx dt + x t = √ t 2 + 1; x = ( t 2 +1) 3 2 + C 3 t 15. D 2 x y- 3 xD x y + 3 y = 2- 3 x 2 ; y = x 2 + 3 x 16. xD 2 x y- xD x y = 12 x 2- 6 x 3 ; y = 2 x 3 + 5 In problems numbered 17 to 40, find the solution of the differential equation satisfying the initial conditions. 17. y = 2 x- 4; y = 3 at x = 3 Solution: y = R 2 x- 4 dx or y = x 2- 4 x + C where C is an arbitrary constant. To determine C substitute in the initial conditions ( x = 3 , y = 3) 3 = 9- 12 + C or C = 6 . The answer is y = x 2- 4 x + 6. Comment: y = x 2- 4 x + C is the general solution. y = x 2- 4 x + 6 is the particular solution. 18. y = 4 x + 7; y = 3 at x = 2 19. y =- x 2- 3; y = 4 at x =- 1 20. ds dt = 3 t- 5; s =- 3 at t = 2 21. F ( t ) = 1- 6 t + t 2 ; F (- 1) = 4 F ( t ) = t 3- 3 t + t + 25 3 22. dv dt =- 2 t + 3; v = 4 at t = 3 23. dy dx = √ 7 x 3 + 3 x- 2; y =- 5 when x = 0 24. y = ax 2 ; y = a when x = 1 25. 1 t dy dt = 2 t 2 + t + 1; y = 5 when t = 1 2 26. ds dt = √ 1- 2 t ; s = 0 when t =- 4 27. 1 x dy dx = 8 3 √ 1 + 2 x 2 ; y = 2 when x = 0 28. y =- x √ 10- x 2 ; y = 3 when x =- 2 29....
View Full Document

## This note was uploaded on 09/22/2011 for the course ECON 101 taught by Professor Mr.tull during the Spring '11 term at De La Salle University.

### Page1 / 12

Differential Equations - Contents 15 Differential Equations...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online