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Consider the Van der Pol oscillator
x′′- µ(1 - x2)x′ + x = 0
(a) Write this equation as a system of first order equations
(b) Taking µ = 2, use MatLab's routine ode45 to compute the solution for the initial value problem x(0) = 0 and x′ (0) = 5 from t = 0 to t = 40. Plot x as a function of t and also plot x vs x′ in phase-space.
(c) Repeat for the initial conditions x(0) = 0 and x′ (0) = 0.01.
(d) What is the ?xed point of the system? Use your numerical results to argue what the long time behavior of the system is.
The Mean Value Theorem for Integrals If f(x) is a continuous function on [a,b] then here is a number c in [a,b] thus, a ∫ b f(x) dx = f(c)(b -a) Proof Let's begin
The hole set of irrational and rational numbers is the set of real numbers and is representing by R. Thus, the real numbers can also be describe in terms of position of a point on
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solve the differential equation dy/dx=f(y)x^n+g(y)x^m by finding a one-parameter group leaving it invariant
A function is an equation for which any x which can be plugged into the equation will yield accurately one y out of the equation. There it is. i.e. the definition of functions w
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Use L''hopital''s rule since lim X-->0 1-cos(x)/1-cos(4x) is in the indeterminate form 0/0 when we apply the limt so by l''hoptital''s rule differentiate the numerator and den
About Zeros in the Denominator of Rational Expressions One thing that you must be careful about when working with rational expressions is that the denominator can never be zero
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