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

1.(a) (i) Eliminate the arbitrary function f from the equation (ii) Eliminate the constants a and b from the equation (b) Show that how to solve, by Jacobi's method, a partial differential equation of the type and illustrate the method by finding a complete integral of the equation (c) (1) Find the general solution of the DE 23% + 20 = (*+ )z. 4.5 (i) Find the general integral of the partial differential equation pq = 1. 3 2.(a) Find the general integral of the partial differential equation through the curve xz = a ,y = 0 7.5 (b) Find the equation of the system of surfaces which cut orthogonally the cones of the system x' + y' +z' = cxy. 7.5 (c) Show that the equation z = px + qy is compatible with any equation f(x, y, z, p, q) = 0 that is homogeneous inx, y and z. Solve completely the simultaneous equations z = px + qy, 2xy(p? + q2) = z(yp + xq). 7.5 (b) Find the particular integral of the partial differential equation (c) Find the general solution of the partial differential equation (b) Solve by Monge's Method 227+ 2xys +y-0. 7.5 (c) Solve using Monge's Method pt- qs = q3. 7.5 5. (a) Using separation of variables, show that the one-dimensional wave equation 22z 122z z(ot) = 0,z(a,t) = 0 for all t has solution Where Ay's and By's are constants. 7.5 (b) Solve the one-dimensional diffusion equation (c) Find the D'Alembert solution of the Cauchy problem for one dimensional wave equation given by a2y 122y y = SInX, at Y= Cosx att= 0,x € R 7.5 6.(a) Determine the solution of the initial boundary value problem (b) Determine the solution of the vibrating string problem hx (c) Derive the equation of traffic flow Where p(x, t) is traffic density, flow rate q depends only on density, i.e.,