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# Partial Differential Equations

### Solving Partial Differential Equations

Please see the attached file for the fully formatted problems. 1. Solve u(0,t) = u(,t) =0 u(x,0) = xsinx ,

### Partial Differential Equations : Characteristic Curves

1.(a) Solve the given equation by the method of characteristic curves. (b) Check your answer by plugging it back into the equation. x &#8706;u/&#8706;x + y &#8706;u/&#8706;y = 0 See attached file for full problem description.

### Partial Differentials Equations : Gradient of the Function

Find the gradient &#8711;g of the function g(x, y) = r^5, where r = sqrt (x^2 +y^2). Hint: introduce a new variable, u = x2 + y2. Express g(x, y) in terms of u and use the chain rule to find dg/dx and dg/dy.

### 8 Calculus Questions : Series, Convergence, Divergence and Partial Sums; Partial and Homogeneous Differential Equations

Please see the attached file for the fully formatted problems.

### Laplace Transforms and Systems of Differential Equations

Solve Problems 1 and 2 using Laplace transforms with Initial Conditions: 1. x'-2y'=1; x'+y-x=0 x(0)=y(0)=0 2. x'+2y'-y=1; 2x'+y=0 x(0)=y(0)=0.

### Solving Differential Equations: Laplace Transforms

Using laplace transforms solve the following problems: 1. Find Y(t) if Y'''' + 2Y'' +Y = 3 + cos 2t with Y(0) = Y'(0) = Y''(0) = Y'''(0) please show the partial fraction expansion and how it was obtained..do not need to invert. replace cos2t with f(t) and show that solution.

### Cauchy partial differential equation

Prove d/dt(u(x(t),t))+tanh(x(t))(d/dx(u(x(t),t)))=0 u(x(t),0)=a(x(t)) limit as x tends to infinity of u(x,t)=0 has at most one solution.Explain why there is no boundary condition at x=0 and find a solution for the special case a(x(t))=sinh(x(t))

### Partial Differential Equations : Separation Solutions

Please see the attached file for the fully formatted problems.

### Cauchy Reimann condition and analytic functions

1. a) The Cauchy-Riemann equation is the name given to the following pair of equations, ∂u/∂x=∂v/∂y and ∂u/∂y= -∂v/∂x which connects the partial derivatives of two functions u(x, y) and v(x, y) i) if u(x, y) =e^x cosy and v(x, y) =e^x siny, how do I prove that these functions satisfy the Cauchy-Riemann equa

### Partial Differential Equation simplification

I am trying to simplify Y(y) using the boundary condition Y(1) = 0. Please find details on the attached file. Thanks

### Cauchy Problem: One-Parameter Family of Solution Curves

I cannot use mathematical symbols. Thus, I will let * denote a partial derivative. For example, u*x means the partial derivative of us with respect to x. Furthermore, I will let u*x=p and u*y=q. Also, I will let ^ denote a power. For example, x^2 means x squared, and, I will let / denote division. Here is my problem: The PDE

### Partial Differential Equations

A) Classify and find general expressions for the characteristic coordinates for the equation {see attachment} b) Use the canonical coordinates {see attachment} and transfer the above PDE into the new coordinates. Solve it in the new coordinates and show that {see attachments} where F and G are arbitrary functions of their ar

### Analyticity : Cauchy-Riemann Equation and Antiderivatives

Let D be a domain in C and assume that f is analytic in D. Decide whether the statements below are true or false and give a short reason for your answer. a) If there exists an open subset U of D such that Im f&#8801;0 in U, then f is constant in D. Please see attached for all three questions.

### Analytic Function : Chain Rule And Cauchy-Riemann Equation

Suppose that a function f(z) is analytic at a point z0 = z(t0) lying on a smooth arc.... Please see the attached file for the fully formatted problems.

### Cauchy-Riemann Equations

6. Let u and v denote the real and imaginary components of the function f defined by the equations _ f(z) = (z^2)/z when z &#8800; 0, f (z) = 0 when z = 0. Verify that the Cauchy-Riemann equations ux = vy and uy = -vx are satisfied at the origin z = (0, 0).

### Partial Differential Equation Problem

Step by step work and explanation and solution. (Solution is provided in the attachment).

### Partial Differential Equations

See Attached Show that satisfies the partial differential equation Note: 1- are constants 2- 3- show does not mean solve the pde

### Greatest Possible Value of Partial Sum

If equation A = 1 what is the greatest possible value of equation B? *(Please see attachment for equations)

### Partial Differential Equations

Suppose that a string of length L is held fixed at one end and is being moved up and down, say with a displacement of f(t), at the other end. Assume that the string is initially at rest and has a zero displacement, and similarly, that the forcing of the string, f(t), also has a zero initial velocity and displacement. a) Write

### Partial Differential Equation (PDE) : Heat Conduction

3. Consider the PDE problem: {see attachment} Suppose v(x,y) represents the temperature of some heat-conducting material. What physical scenario could be described by this PDE problem? What does each equation mean physically? Solve for v(x,y). Your final answer should indicate how all constants are obtained from g(x).

### Partial Differential Example Equations

Consider a thin rod of heat-conducting material with length L. Suppose that the rod is initially heated to a temperature of T uniformly throughout the tod, and is dropped into a bucket of ice water at t = 0. Suppose that the rod is everywhere insulated, except for its left end (x = 0), which is expoosed to the ice water. (a)

### Partial Length of Chord

I have a circle, diameter and radius unknown. It has a chord running through the center (point "O") and the end points are on the circle. Lets call this chord "SR" This chord SR bisects another chord "TU" and TU has midpoint "K". The only other information I have is that arc angle "TR" formed by point "T" from chord TU and point

### Linear/Non-Linear Differential Equation

See attachment.

### Mathematical Modelling : Elliptic Partial Differential Equation

Please see the attached file for the fully formatted problem. Show that the PDE ... is of elliptic type for 0<y<1and all x does not equal 0; what is it for y>1and y<0 (and all x does not equal 0)?

### Impulse Forcing

You have a mass-spring system, a unit impulse is applied to this system (at equilibrium,at rest) and the response is recorded and determined to be (10e^-0.1t)- (10e^-0.2t) In general terms what does the form of the impulse response function tell you about the system?

### Non-Homogeneous Linear Partial Differential Equation

Linear Partial Differential Equation (II) Non- Homogeneous Linear Partial Differential Equation with Constant Coefficients Problem: Find the solution of the equation (D2 - D'2 + D -

### It is an explanation for solving homogeneous linear partial differential equation. Find the General solutions of the equations r = a2t and (D + D')z = sin x.

Linear Partial Differential Equation (I) Linear Homogeneous Partial Differential Equation with Constant Coefficients Problem 1: Find the General solution of the equation r = a2t.

### Laplace Transforms : Convolution Products and Differential

Please see the attached file for the fully formatted problems. LAPLACE TRANSFORMS (1) Calculate the following convolution products :- t t -t (a) t * e