1. Solve the following differential equations (5×4 = 20)(a) xdy ydx x y dx 2 2 − = −(b) ) 01) (31(4 3 3 4 2 + + − dy =ydx x yxx y(c) y x xdxd y+ = + 222(d) y x xdxdydxd y4 4 4cos 3 sin22+ + = +2. A mass of 60 kg slides on a table. The frictional force is 60 times the velocity and the mass isbeing pushed with a force of 54 sin 2t N. Find the velocity as a function of time if v = 0 att = 0. (10)3. A circuit consists of an inductance of 0.05 H, a resistance of 20W, a capacitor of 100 μF andan emf of 100 cos 200 t. Find i and q given that q = 0 and i = 0 at t = 0. (10)4. Obtain the power series solution for the following differential equation:(1 ) 2 2 0222 − − + y =dxdyxdxd yx (10)5. Show that the function (10×2 = 20)a) u = x4 − 6x2 y2 + y is a solution of the two-dimensional Laplace equation.b) u = sin w ct sin wx is a solution of the one-dimensional wave equation.6. Write down the equationu t c u 2 2 2 2 ¶ / ¶ = Ñin plane polar coordinates. By the method of separation of variables, reduce the PDE soobtained to a set of ODEs. (10)7. Represent the following function by a Fourier sine series/ 8 / 40 / 8if4if( )p < < p< < p

−= pttttf t (10)8. Find the solution of the wave equation for a vibrating string of length l = 10 units with endsfixed. It is given that c2 = 1, initial velocity is zero and initial deflection isu (x, 0) = k (sin x − sin 2x)

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