1. Use the separation of variables, u(r, 0) = R(r)e(0), to solve the following partial differential equation1 au+drJuV?u =r Ər(1)in a domain with 1

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Answered: 1. Use the separation of variables,…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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A. Classify the given DE according to order, degree and type Differential Equation Order Degree Туре 1. (2xy + x²)dx + 2ydy = 0 d³y, 2. ()3 - x()² + x² = 0 'dx³- 'dx²' 3. (x - 1)²y" + xy' + y = 0 əx 'Xze' 4. + 5y(2 = 8 ду ду? d³y 5. dx³ x2 - 3x = 4
Consider the following partial differential equation: y(e +1)u, – e"(y² + 1)²u, = 0, y >0 1. By using the separation method with u(r, y) = f(r)+ g(y) to solve (F), we obtain that : (F) 1+ e" df (y² + 1)² dg a) da fip df b) 1+ er dr (y? + 1)? dg dy dg et %3D 1+ e" df c) et dr (y? +1)2 dy d) None of the above. 2. A solution u(x, y) of (F) is ( B is an arbitrary constant): a) u(r, y) = À (In(1+ e*) – In(y² +1)) + B. 1 b) u(x, y) = (In(1 + e")- + B. 2(y² + 1) 1 c) u(r, y) = A + B. 2(y? + 1). d) None af the above.
3. Find a function y= f(x) satisfying the given Differential Equation and the prescribed initial condition a) dy - Jx,y(4) = 0 dx b) -xVx* +9,y(-4)-0 dx dy c) dx x +1 10 y(0) = 0 %3D
6. Use separation of variables to find product solution of u(x, y) for the partial differential equation a?u du =-+ xu ay? ax
Given the partial differential equation Ut + Uxx = 0,0 0. Use the method of separation of variables to solve the given equation with the following conditions. Ux(0, t) = 0,t > 0 Ux(T, t) = 0, t > 0 U(x, 0) = x(1 – x), 0 < x < n
4. Consider the autonomous ODE, y' = f(y) = -y(y² – y – 6). (a) Identify the equilibrium solutions and classify the stability of each. d²y directly dt? (b) Identify points of inflection for the solution curves by finding an expression for from the autonomous differential equation. d to the equation dt dy = f(y).) dt (Apply - (c) Sketch a few solution curves in the (t, y) plane, including the location of inflection points.
Use the method of separation of variables to find the general solution of the following partial differential equation du Ju Ət - u subject to the initial condition ?х u(x,0) = ex

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1. Use the separation of variables, u(r, 0) = R(r)e(0), to solve the following partial differential equation 18 du] 10²u rər Ər 7-²00² T + (1) in a domain with 1