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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 CONTINUOUS TIME:LINEAR DIFFERENTIAL EQUATIONS Economic Applications LESSON 2 prof. Beatrice Venturi Mathematics for Economics Beatrice Venturi
CONTINUOUS TIME : LINEAR ORDINARY DIFFERENTIAL EQUATIONS ECONOMIC APPLICATIONS Mathematics for Economics Beatrice Venturi
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
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
LINEAR FIRST ORDER DIFFERENTIAL EQUATIONS (E.D.O.) Mathematics for Economics Beatrice Venturi
LINEAR FIRST ORDER DIFFERENTIAL EQUATIONS (E.D.O.) GENERAL SOLUTIONOF (1) Mathematics for Economics Beatrice Venturi
FIRST-ORDER LINEAR E. D. O. Example Mathematics for Economics Beatrice Venturi
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
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
FIRST-ORDER LINEAR E. D. O. • [Plot] Mathematics for Economics Beatrice Venturi
The Domar Model Mathematics for Economics Beatrice Venturi
The Domar Model • Where s(t) is a t function Mathematics for Economics Beatrice Venturi
LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS • The homogeneous case: Mathematics for Economics Beatrice Venturi
LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS Separate variable the to variable y and x: We get: Mathematics for Economics Beatrice Venturi
LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS We should able to write the solution of (1). Mathematics for Economics Beatrice Venturi
LINEAR FIRST ORDER DIFFERENTIAL EQUATIONS (E.D.O.) 2) Non homogeneous Case : Mathematics for Economics Beatrice Venturi
CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS • We have two cases: • homogeneous; • non omogeneous. Mathematics for Economics Beatrice Venturi
CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS : a)Non homogeneous case with constant coefficients b)Homogeneous case with constant coefficients Mathematics for Economics Beatrice Venturi
CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS We adopt the trial solution: Mathematics for Economics Beatrice Venturi
CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS We get: This equation is known as characteristic equation Mathematics for Economics Beatrice Venturi
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
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
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
CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS • Examples The complentary function: The solution of its reduced homogeneous equation Mathematics for Economics Beatrice Venturi
CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS Mathematics for Economics Beatrice Venturi
CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS Mathematics for Economics Beatrice Venturi
CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS • The particular solution:: The General solution Mathematics for Economics Beatrice Venturi
CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS The Cauchy Problem Mathematics for Economics Beatrice Venturi
CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS x(t)= Mathematics for Economics Beatrice Venturi
CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS Mathematics for Economics Beatrice Venturi
CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS Mathematics for Economics Beatrice Venturi
CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS Mathematics for Economics Beatrice Venturi