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... . So if m is even, we find solutions for x2 − ny 2 = 1 with every turn through the period; if m is odd, we find solutions with every two turns (and solutions to x2 − ny 2 = −1 for the alternate turns). ...
... . So if m is even, we find solutions for x2 − ny 2 = 1 with every turn through the period; if m is odd, we find solutions with every two turns (and solutions to x2 − ny 2 = −1 for the alternate turns). ...
![[2014 solutions]](http://s1.studyres.com/store/data/008881817_1-9b59b3a488861f750cad1da068f01a69-300x300.png)
[2014 solutions]
... (b) It is a double well potential, symmetric about x = 0 with a local maximum at x = 0 (where V = ga4 ) and global minima at x = ±a, where V = 0. As |x| → ∞, the potential V (x) → ∞. (c) L = T − V = 12 mẋ2 − g(x2 − a2 )2 . Equation of motion is the Euler-Lagrange equation d ∂L = ∂L or mẍ = −4gx(x2 ...
... (b) It is a double well potential, symmetric about x = 0 with a local maximum at x = 0 (where V = ga4 ) and global minima at x = ±a, where V = 0. As |x| → ∞, the potential V (x) → ∞. (c) L = T − V = 12 mẋ2 − g(x2 − a2 )2 . Equation of motion is the Euler-Lagrange equation d ∂L = ∂L or mẍ = −4gx(x2 ...
Slide 1
... solutions. Solving Using a Given Domain – Start with an equation and a set of numbers for one variable only. You then substitute each number in for the variable it replaces, and solve for the unknown variable. This gives you a set of ordered pairs that are solutions. ...
... solutions. Solving Using a Given Domain – Start with an equation and a set of numbers for one variable only. You then substitute each number in for the variable it replaces, and solve for the unknown variable. This gives you a set of ordered pairs that are solutions. ...
H5
... 3. To practice the idea that we can work with any field in the same way that we work with Q, R, or C, let’s do some linear algebra in Z/pZ where p is a prime. ...
... 3. To practice the idea that we can work with any field in the same way that we work with Q, R, or C, let’s do some linear algebra in Z/pZ where p is a prime. ...
PS9
... This u(t, x) should actually be the sum of two expressions, one which satisfies the unforced (i.e., homogeneous) wave equation, and the other which satisfies the forced one. Identify this latter part, and compare it to the forcing function f (t, x) itself. (c) Calculate an expression for u(t, x) aga ...
... This u(t, x) should actually be the sum of two expressions, one which satisfies the unforced (i.e., homogeneous) wave equation, and the other which satisfies the forced one. Identify this latter part, and compare it to the forcing function f (t, x) itself. (c) Calculate an expression for u(t, x) aga ...
Partial differential equation
In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives. (A special case are ordinary differential equations (ODEs), which deal with functions of a single variable and their derivatives.) PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a relevant computer model.PDEs can be used to describe a wide variety of phenomena such as sound, heat, electrostatics, electrodynamics, fluid flow, elasticity, or quantum mechanics. These seemingly distinct physical phenomena can be formalised similarly in terms of PDEs. Just as ordinary differential equations often model one-dimensional dynamical systems, partial differential equations often model multidimensional systems. PDEs find their generalisation in stochastic partial differential equations.