Partial Differential Equations — DU B.Sc. (Hons.) Mathematics Previous Year Question Papers

1. (a) Define characteristic curves of a first order partial differential equation. Find the partial differential equation arising from the two-parameter. family of curves: (b) Determine the integral surfaces of the equation: xy? + 4)u, - y(x? + u)U, = (x2 -y')u; with the data x + y = 0 and u = 1. (c) Find the solution of the initial value system: 2. (a) Reduce the given equation into canonical form and find the general solution x: (b) Using v = In u and v(x,y) = f(x) + g(y), solve the equation: (c) Find a complete integral of the following equation by using Charpit's method: 3. (a) Derive the three-dimensional Laplace equation: 0x=x03y ay? + az? = 0, where V is Gravitational potential at any point. (b) Determine the region of the following equation, and transform it to canonical form: (c) Derive the equation of traffic flow: Where p(x, t) is traffic density and q(x, t) is traffic flow at any time t. 4. (a) Find the general solution of the equation: (b) Reduce the following equation into canonical form : "xx- 2cox Wxy + (1 + cos)u, + u = 0. (c) Find the general solution of the equation: 5. (a) Determine the solution of the initial-value problem for an infinite string with initial condition: Utt-Ux = 0, x ER, t > 0, u (x, 0) = sin3x, x E R, (b) Find the solution of the initial-value problem for a semi-infinite string with fixed end: (c) Determine the solution of wave equation of semi-infinite string with free end for non- homogeneous boundary value problem: 6. (a) Obtain the solution for a semi-infinite string with free end: (b) Determine the solution of wave equation for non-homogeneous boundary value problem of semi-infinite string with fixed end: (c) Determine the solution of Cauchy problem for non-homogeneous wave equation: Utt = Uxx - 2, xE. R, t > 0, with initial condition u(x, 0) = sin x, u,(x, 0) = x,