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AMATH 250 (22)


46 Pages

Applied Mathematics
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Julia Roberts

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Introduction to Differential Equations (DE's) September-10-12 2:01 PM Algebraic equation: Two numbers make this an identity DE for unknown function - t is independent variable - y is dependent variable where and when substitute into DE, we get identities for Example: Newton's Second Law is force, may depend on For vertical motion of small object of mass , known Free fall: Solve the DE Solve by using antidifferentiation This is a two-parameter family of solutions State of motion in mechanics is given by and Given initialconditions at : and what is at ? AMATH 251 Page 1 Modeling with DE's September-12-12 Example: Free Fall 1:39 PM is a given constant/parameter General Solution to DE Contains arbitrary constants General Solution Initial Conditions (IC) and are arbitrary constants and they make this a general solution What it sounds like: initial conditions that allow the general It is a two-parameter family of solutions solution to the DE to be fixed to a single result Autonomous DE Initial Conditions (IC's) at The independent variable does not appear. In this situation we can introduce a new dependent variable Look at the general solution and find and in general solution to accommodate the IC's for , where is the independent variable. Classification of DE's For some (*) is an order differential Particular Solution equation. Generally non-linear which satisfies both the original differential equation and the initial conditions. As such, it predicts at any and at We solved an Initial Value Problem (IVP) — a DE plus IC's Linear Differential Equation Drag Coefficients are given functions of as is the term Generalization of the model for motion in a field of gravity (2nd law of motion) For , this is anear, non-homogeneous, order a) Differential Equation is the friction/drag coefficient —always opposes motion If then the DE is calledhomogeneous. Changing Gravity If is "large", force of gravity is given by b) This equation is calledautonomous since the independent variable does not appear, only its derivatives. For a) Introduce new dependent variable and write So we now have a system of two first order DE's For b) Rocket burning fuel Rocket burning fuel, expelled at at velocity relative to rocket Mass of rocket: AMATH 251 Page 2 Solutions Explicit Solution Example September-14-12 1:34 PM Show that (*) is a general solution for Explicit Solution A function that, when substituted for in the DE (*), satisfies this equation for all is called anexplicit solution. Implicit Solution Example Implicit Solution Show that relation for some positive constant is the implicit solution to A relation , for some is said to be an implicit solution to the differential equation for , for some on interval if it Use implicit differentiation assumingy is a function of x defines one or more explicit solutions on Solutions of order Differential Equations (*) for some defines a circle so there are two explicit solutions: Subject to the initial condition for The Existence and Uniqueness Theorem Initial-Value problem for order DE for the Initial Value Problem (IVP) (*) If and are continuous functions on General solution that contains the point In general, there will be integration constants then the IVP (*) has a unique solution in some interval Find to accommodate initial conditions: Example Does the DE have a unique solution such that a) Yes, for b) No unique solution. Both are solutions General solution Solution Methods Direction Fields for 1st order DE's , we can generate a picture of the family of solution curves that correspond to general solutions. Look for isoclines with constant slope Example Rays exiting from the circle are isoclines. Autonomous Equations For autonomous DE 1) Look for equilibriumsolutions such that . Solve Example AMATH 251 Page 3 Example Euler's Method September-17-12 Euler's Method for 1:34 PM Euler's Exact st 0 1 4 4 Numerical Approximation for 1 -order DE's Euler's Method 1 1.1 4.200 4.213 2 1.2 4.425 4.452 3 1.3 4.678 4.720 Find approximationfor exact solution 4 1.4 4.959 5.018 5 1.5 5.271 5.318 Use partitionof t-axis Example Linear approximation: Evaluate using Euler's method Exact: Approximation: 1 1 2 2 0.5 2.250 4 0.25 2.441 8 0.125 2.566 16 0.0625 2.638 Dimensions of PhysicalQuantities Free Fall September-17-12 2:02 PM In mechanics, all physical quantities have dimensions of form ← M ss ← L ← Time Consistency Requirements 1) Dimensional Homogeneity of equation: Work May only add, subtract or equate quantities that have the same dimension. 2) Quantities havingdifferent dimensions may only be combined by multiplicationof division Notation for quantity has dimension Free fall with Drag Dimension of drag coefficient Dimensionless time Using dimensionless time: notice Define dimensionless time: Solving First Order DEs September-19-12 Separable DE 1:25 PM Let st Types of 1 Order DE's That can be solved analytically L Solution is given by 1) Separable Equation In general this is implicit 2) Linear Equation Example 3) Exact Equation Example Recall example of a body experiencing gravity and drag force Define dimensionless time Define terminal velocity so that Define Stable equilibrium solution where Population Growth y = number of species at time with Simple equation of growth with rate K is the carrying capacity of the system This is an autonomous equation, there is no in Equilibrium solution: Stable equilibriums have unstable have Solve by separable variables Partial fraction decomposition AMATH 251 Page 6 Partial fraction decomposition s where Initial Condition Second Order Des with missing independent variable s Let and use chain rule. Chain rule: In mechanics, separable DE AMATH 251 Page 7 More Solving In mechanics, Newton in 1-D September-21-12 , missing independent variable t 1:37 PM Existence and Uniqueness y "independent" Theorem s and If and are continuous in interval I containing the initial point then there exists a unique solution to IVP for all Conservation of energy IC at Solve for Sub into 1 Solve as separable DE Escape velocity Minimum speed to send rocket to space escape velocity: s Cases of DE's that may be converted to separable DE's Homogeneous 1) Example Use substitution Assignment Q1 2) Define Linear Equations given function of t IC Solve by Method of Integrating Factors (see problem 39 in Sec.1.2, page 26 for Method of Variation of Parameters) Bernoulli's equation Bernoulli's equation Define Assignment Q2 Exact Solution Proof Exact Differential Equations September-24-12 If is a solution to (*) then upon substitution into , use Chain rule 1:28 PM Exact Equation (*) This differential equation is called an exact equation if there exists a that is such that Proof of Test of Exactness Its general solution is given by Show f exact, then there exist F, and Differentiate Test of Exactness Let be continuous functions on a simply-connected domain in Since and are continuous Then is an exact equation if and only if in Other way Assume and Integrate with respect to , keep fixed. Define such that Use this to determine This must be independent of , derivative with respect to must vanish. Example Solve the differentiation equation Test for exactness Integrate Putting together Initial condition: gives AMATH 251 Page 10 Second Order Differential Equations Modelling with 2nd -order DE's with constant coefficients September-24-12 Mechanical sprint -mass oscillator 2:10 PM Standard Form of Linear (Non-HomogeneousDE) Giventhethreefunctions of Standard Form of Linear With Constant Coefficients external force are constant restoring force of spring drag force Origin is the equilibrium position (unstretched spring) s s Displacementofthe mass fromequilibriumpositionfollows Newton s Linearization of Pendulum Linear Displacement Forcesactingtangentially s s B s s s Forsmallangulardisplacements: s Dropall powersbut theleadingone s Initial Conditions Electrical Oscillator: Series RLC Circuit AMATH 251 Page 11 s s State of the RLC circuit is defined by s s s Kirchhoff's Voltage Law Initial Conditions Equivalent DE for Current Differentiate Initial Conditions To get , set in AMATH 251 Page 12 How to represent every (general) solution to 2 ndorder homogeneous linear DE Theory of order linear DE's September-26-12 A) Need and linearly independent 2:08 PM Forget solutions to (*) which are identically zero on (trivial solutions) Note If is a complex-valued solution of DE (*) with real-valued coefficients, Initial Conditions so are its real , and imaginary parts. Existence and Uniqueness Theorem Proof of Theorem (Solution to IVP) If and are continuous functions on some interval , which contains , then there exists a unique solution to the differential equation for all ; which satisfies the initial conditions. Linear Operator Define linear operator ← s ← H s Proof of Theorem 1) Assume Theorem If and are sufficiently differentiable on the interval and 2) . Assume that for some and are any constants then Corollary (Superposition Principle) s If and are any solutions of the homogeneous DE then the linear combination is also a solution to that same DE. Proof of Abel's Theorem Linear Independence For Two functions and are said to be linearly independent on interval iff neither is a constant multiple of the other. If and are solutions to then Fundamental Set (Basis) If and are solutions of the homogeneous DE (*) that are linearly is given by independent on then they are said to form a fundamental set, or basis, of solutions. Proof Theorem: Representation of General Solution to (*) Representation of general solutions to homogeneous, linear, second- order differential equation. If and are linearly independent solutions of linear on , then every solution of that equation is give by where and are arbitrary constants that may be determined from the Comment are arbitrary. Wronskian The Wronskian of two functions and is defined by Abel's Theorem for Wronskian For any two solutions of the DE (*), and then Corollary is either never zero or always zero on so the value of depends on point Theorem If and are any two solutions of on , then they are linearly dependent on iff their Wronskian is identically zero on . AMATH 251 Page 13 Example Solving order DEs with constant coefficients Solve the IVP for October-01-12 1:32 PM Characteristic equation General solution: Fundamental Solutions Find the fundamental solutions to s s and Assume solution in form with being a parameter and the solution is Solve the characteristic equation (auxiliary equation): Example These two roots give us These are linearly independent iff 1) Distinct real roots, s s 2) Distinct complex roots s s s s s s s s s 3) Equal roots, We only have Use reduction of order, assuming s s Alternate method How can we write this as Assume and let s General solution: s s s s s Expand with Maclaurin series s s s and depend on so that s Example s Conclusion If repeats the just multiply by a single factor of Non-Homogeneous Linear Equation Let be general solution of associated linear equation: e.g. and let be any particular solution of the non-homogeneous equation Then the general solution of the equation is AMATH 251 Page 14 Superposition Principle for Non-Homogenous Linear DEs October-03-12 1:26 PM Example General Solution to Linear Second Order Non-Homogeneous DEs Find to Let are undetermined coefficients general solution of associated homogeneous equation s Since and are linearly independent functions Then the general solution of is Comment If and and solve and Example then solves Method of Undetermined Coefficients DE with constant coefficient s If is • polynomial in • exponential Example • s or s • or product of the above Use Method of Undetermined Coefficients 1) , assume What went wrong? 2) assume Recall: linearly independent solutions of the associated homogeneous DE 3) s or s assume were For , assume Exception If reproduces any of the functions in the basis of solutions to the homogeneous DE, then just multiply your assumption for by a single factor of . Method of Variation of Parameters (or Constants) for Example If we have a fundamental set of solutions Solve IVP for Recall To find particular solution Assume Need to determine functions Assume Sub into initial DE 1) 2) Example s Recall Example s s s s s Quiescent Initial Conditions s s Compare solutions to the IVPs 1) s s s 2) s s Difference is solutions of the IVP AMATH 251 Page 15 2) s s Difference is solutions of the IVP With 0 for initial conditions, called Quiescent initial conditions. Example of Method of Variation of Parameters Find for s From s s s s s s s s s Solution to the IVP with Quiescent initial conditions. s Integrate formulas for such that s s s s s s s s Example - Multiplication by t for constant coefficients Find for Green's Propagator Recall Use method of variation of parameters serves in some sense as an inverse of but is linearly dependent on so AMATH 251 Page 16 OscillatorDE and Resonance October-10-12 1:31 PM Recall: mechanical and electrical oscillators • Mass on spring, possiblydamped • RLC circuit Both systems are describedby where or = natural frequency = damping parameter. Normally Assume harmonicforcing s F = amplitude, = frequency Free Oscillations First consider free oscillationswith The associatedhomogeneousDE Aside: we could use dimensionlesstime Determinantcases: 1) : overdamped motion of oscillator 2) : critically damped oscillator 3) : underdamped oscillator s s s and are determined by initial conditions: Forced Oscillations s General solution: s s s s Undeterminedamplitude, phase shift Steady-StateResponse Since persists, is called steady-stateresponse. Transient Response Since , it is called the transientresponseof the system. The initialconditionsare "forgotten". Solving To determine and assume complex-valued solution s AMATH 251 Page 17 s s Find particular solution and take Assume where s Resonance Amplitude s s s s Solution to Forced Oscillations s Remark: Notation in textbook Analyze this as function of "reduced" frequency Resonance For , there is a local maximum in the plot of . This is the resonantfrequency. If Zero Damping s s s Persistsfor all t s s s Recall s Resonance Phase General solution s What happens as s "Borrow" part from and consider quiescentsystem s s s s s s Beats consistsof a large amplitude and large period sine wave filled with a s small amplitude wave. These are known as beats s s This is a linearly-increasingamplitude sine wave. AMATH 251 Page 18 s This is a linearly-increasingamplitude sine wave. AMATH 251 Page 19 Systems of First Order DEs October-15-12 Homogeneous Undamped Oscillator 1:55 PM Solve IVM for homogeneousequation Phase Portrait Characteristic equation The phase portrait of the solution is the description of a s s solution to the DE as a circle on theand plane. s s s s s s The vector defines the state of the system at time s s s s Describe the state for the system by a curve in the plane. Rewrite as a system of equations for state variables Define the vector so the system becomes This looks like a separable equation Matrix exponential: s s s s s s where s This is a circle on the plane. Phase portrait Recall Multiply by and obtain This equation represents conservation of energy. Laplace Transform ConsidertheoscillatorDE periodicforcing October-17-12 1:38 PM Example: Integral Transform An IntegralTransformis a linear operatorthat maps functions tofunctions , definedby is called the Kernel Laplace Transform Laplace transform of , defined on is the function defined by How do we solve this? 1) Undetermined coefficients forsolutionsoverintervalswhen is continuous Note 2) The Green'sfunction In thiscourse, is realbut in general it is possiblethat What about higher-order linear DE's with constant coefficients ? Notation These arise with coupled oscillators. The domain of definition of is the set of alvalues for 3) Can use undeterminedcoefficients which the integral exists (converges). Note:since involvesan improperintegral This has complex roo
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