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Economics Faculty

Economics Faculty. CONTINUOUS TIME: LINEAR DIFFERENTIAL EQUATIONS Economic Applications. LESSON 2 prof. Beatrice Venturi. CONTINUOUS TIME : LINEAR ORDINARY DIFFERENTIAL EQUATIONS ECONOMIC APPLICATIONS. LINEAR FIRST ORDER DIFFERENTIAL EQUATIONS (E.D.O.). Where f(x) is not a constant.

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Economics Faculty

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  1. Economics Faculty CONTINUOUS TIME:LINEAR DIFFERENTIAL EQUATIONS Economic Applications LESSON 2 prof. Beatrice Venturi Mathematics for Economics Beatrice Venturi

  2. CONTINUOUS TIME : LINEAR ORDINARY DIFFERENTIAL EQUATIONS ECONOMIC APPLICATIONS Mathematics for Economics Beatrice Venturi

  3. LINEAR FIRST ORDER DIFFERENTIAL EQUATIONS (E.D.O.) Where f(x) is not a constant. In this case the solution has the form: Mathematics for Economics Beatrice Venturi

  4. LINEAR FIRST ORDER DIFFERENTIAL EQUATIONS (E.D.O.) We use the method of integrating factor and multiply by the factor: Mathematics for Economics Beatrice Venturi

  5. LINEAR FIRST ORDER DIFFERENTIAL EQUATIONS (E.D.O.) Mathematics for Economics Beatrice Venturi

  6. LINEAR FIRST ORDER DIFFERENTIAL EQUATIONS (E.D.O.) GENERAL SOLUTIONOF (1) Mathematics for Economics Beatrice Venturi

  7. FIRST-ORDER LINEAR E. D. O. Example Mathematics for Economics Beatrice Venturi

  8. FIRST-ORDER LINEAR E. D. O. We consider the solution when we assign an initial condition: y′-xy=0 y(0)=1 Mathematics for Economics Beatrice Venturi

  9. FIRST-ORDER LINEAR E. D. O. When any particular value is substituted for C; the solution became a particular solution: The y(0) is the only value that can make the solution satisfy the initial condition. In our case y(0)=1 Mathematics for Economics Beatrice Venturi

  10. FIRST-ORDER LINEAR E. D. O. • [Plot] Mathematics for Economics Beatrice Venturi

  11. The Domar Model Mathematics for Economics Beatrice Venturi

  12. The Domar Model • Where s(t) is a t function Mathematics for Economics Beatrice Venturi

  13. LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS • The homogeneous case: Mathematics for Economics Beatrice Venturi

  14. LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS Separate variable the to variable y and x: We get: Mathematics for Economics Beatrice Venturi

  15. LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS We should able to write the solution of (1). Mathematics for Economics Beatrice Venturi

  16. LINEAR FIRST ORDER DIFFERENTIAL EQUATIONS (E.D.O.) 2) Non homogeneous Case : Mathematics for Economics Beatrice Venturi

  17. CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS • We have two cases: • homogeneous; • non omogeneous. Mathematics for Economics Beatrice Venturi

  18. CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS : a)Non homogeneous case with constant coefficients b)Homogeneous case with constant coefficients Mathematics for Economics Beatrice Venturi

  19. CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS We adopt the trial solution: Mathematics for Economics Beatrice Venturi

  20. CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS We get: This equation is known as characteristic equation Mathematics for Economics Beatrice Venturi

  21. CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS Case a) : We have two different roots The complentary function: the general solution of its reduced homogeneous equation is where are two arbitrary function. Mathematics for Economics Beatrice Venturi

  22. CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS Caso b)We have two equal roots The complentary function: the general solution of its reduced homogeneous equation is dove sono due costanti arbitrarie Mathematics for Economics Beatrice Venturi

  23. CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS Case c) We have two complex conjugate roots , The complentary function: the general solution of its reduced homogeneous equation is This expession came from the Eulero Theorem Mathematics for Economics Beatrice Venturi

  24. CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS • Examples The solution of its reduced homogeneous equation Mathematics for Economics Beatrice Venturi

  25. A solution of the Non-homogeneous Equation • A good technique to use to find a solution of a non-homogeneous equation is to try a linear combination of  a0(t) and its first and second derivatives. Mathematics for Economics Beatrice Venturi

  26. A solution of the Non-homogeneous Equation • If,   a0(t) = 6t3 -3t , then try to find values of A, B, C and D such that A + Bt + Ct2 + Dt3is a solution. Mathematics for Economics Beatrice Venturi

  27. The solution of the Non-homogeneous Equation • Or if   • f (t) = 2sin t + cos t, then try to find values of A and B such that • f (t) = Asin t + Bcos t is a solution. • Or if • f (t) = 2eBt • for some value of B, then try to find a value of A such that AeBt is a solution. Mathematics for Economics Beatrice Venturi

  28. A solution of the Non-homogeneous Equation We consider for example: The function on the right-hand side is a third-degree polynomial, so to find a solution of the equation, we have to try a general third-degree polynomial, that is, a function of the form: =A + Bt + Ct2 + Dt3. Mathematics for Economics Beatrice Venturi

  29. CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS Mathematics for Economics Beatrice Venturi

  30. CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS Mathematics for Economics Beatrice Venturi

  31. CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS • The particular solution is: Thus the General solution of the original equation is Mathematics for Economics Beatrice Venturi

  32. CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS The Cauchy Problem Mathematics for Economics Beatrice Venturi

  33. CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS x(t)= Mathematics for Economics Beatrice Venturi

  34. CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS Mathematics for Economics Beatrice Venturi

  35. CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS Mathematics for Economics Beatrice Venturi

  36. CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS Mathematics for Economics Beatrice Venturi

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