Consider the following partial differential equation. au, au ax? + = u ay? Identify A, B,...
Consider the following partial differential equation. 22²ua²u at? ax² Identify A, B, and C in the above equation and use them to calculate the following. B2 - 4AC = x Classify the given partial differential equation as hyperbolic, parabolic, or elliptic. hyperbolic parabolic elliptic
b) i. Form partial differential equation from z = ax - 4y+b [4 marks] a +1 ii. Solve the partial differential equation 18xy2 + sin(2x - y) = 0 дх2ду c) i. Solve the Lagrange equation [4 Marks] az -zp + xzq = y2 where p az and q = ду [5 Marks] x ax ii. A special form of the second order partial differential equation of the function u of the two independent variables x and t is given...
(3 points) For the partial differential equation –7824 au aray - 7 - 9 – 7u = 0 the discriminant is ar2 The PDE is (check all that apply) A. hyperbolic B. parabolic c. elliptic D. separable E. inseparable
4. (50 pts) Consider the following partial differential equation: 1du au Ət22 Ətər2 (BC) u7,t) = 0 20,t) = 0 0 <t (IC) u(3,0) = 0 0 <r <a Follow the steps below to solve it: (a) (8 pts) Separate variables as u(x,t) = X(2)T(t) to derive the following differential equations for X and T, with an unknown parameter 1: T" - T' + XT = 0, X" + 1X = 0.
9) Solve the following partial differential equation au a2u ax2 n(0, t) = u(2, t)-0 t > 0 (x, 0) = 0 au at It=0 =x(2-x) 9) Solve the following partial differential equation au a2u ax2 n(0, t) = u(2, t)-0 t > 0 (x, 0) = 0 au at It=0 =x(2-x)
Consider the following nonhomogeneous linear differential equation ay 6) + by(s) + cy!4) + dy'"' + ky'' + my' + ny=3x²3x - 7cos +1 where coefficients a, b, c, d, k, m, n are constant. Assume that the general solution of the associated homogeneous linear differential equation is YAEC,+Ce**+ c xe** + c.xe3* + ecos What is the correct form of the particular solution y of given nonhomogeneous linear differential equation? Yanitiniz: o Yo=Ax*e** + Ex + F **+Cxcos() +oxsin()+Ex+F...
Consider the partial differential equation for the function y(x, y) ay ay Әх ду? - ryu = 0 (i) State whether this equation is linear homogeneous, linear inhomogeneous, or non-linear. Justify your statement. (ii) Separate the variables in this equation. Find the separate equations for the variables x and y. (iii) Find the general solutions for each of the separated equations.
Please help me with this ! Thank you 4. Consider the partial differential equation au au at + 0 Əx a) An explicit finite difference representation if this equation can be written as Ui,n+1 = Ui,n [Ui+1,1 – Ui-1,1] where 2 Perform a Von Neumann stability analysis to show that this finite difference equation is unconditioanlly unstable. Δt Δx b) By replacing the term Ui,n by the average of its neighboring values at the same time step, the following explicit...
2. Consider the following partial differential equation (a) Separate this equation into two ordinary differential equations (b) Translate the following boundary conditions on the above partial differential equation to conditions on the ordinary differential equations found above. 2. Consider the following partial differential equation (a) Separate this equation into two ordinary differential equations (b) Translate the following boundary conditions on the above partial differential equation to conditions on the ordinary differential equations found above.
Consider the differential equation for the vector-valued function x, x = x, A- Find the eigenvalues A, , and their corresponding eigenvectors V, V, of the coefficient matrix A (a) Eigenvalues Au, dy (b) Eigenvector for A, you entered above (c) Eigenvector for A, you entered above: V2 = (d) Use the eigenpairs you found in parts (a)-(C) to find real-valued fundamental solutions to the differential equation above X = X Note: To enter the vector (u, v) type <u,v>...