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l. Solve the following equations. In part b), express the general solution in terms of appropriate parameters. a) cos3 (x) tanh(y)dy - sin(x)dx = 0, y(O) = O; b) d��t) + 2 d 2 d ��t ) + x(t) = 0.2. a) Compute .C(eLet .C denote 2 the Laplace transform. t sin(1rt)). b) Prove that .C (it f(u)du) = }.c(f). c) Compute the Laplace transform of J(t) = { s � n t, t ) d) Compute .C(t sin(t)). 0 � t < 21r; t � 21r. 3. Consider the family F of circles given by F : x 2 + (y - c ) 2 = c 2 , c E R a) Write down an ODE y' = F(x, y) which defines the direction field of the trajectories in F. Draw a sketch. b) Write down an ODE which defines a direction field perpendicular to the one you found in part�- That is, find a direction field whose slope at (x, y) in the phase plane is orthogonal to the slope given by F(x, y). Draw a sketch. [Hint. Use the fact that if y1 and y2 are orthogonal curves, then at the point of intersection: dy1 . dy2 = - 1.] dx dx c) Find the curve through (1, 1) which meets every circle in the family F at an angle of 90° . Draw a sketch. [ Hint. Recall that the angle of intersection between two curves is defined as the angle between their tangent lines at the point of intersection.]4. Consider the linear 2nd order ODE: (b) y" - 2y' + 2y = R(t). Solve (b) in the following two cases, with R(t) given below: a) R(t) = O; (give the general solution) b) R(t) = et sec(t). (give a particular solution)5. (10 pts) Solve the IVP: y" + 2y' + 2y = 821r (t), y(0) = 0, y'(0) = 0, t > 0. (Here, 821r is the Dirac distribution at 21r.) Where is your solution discontinuous? Where is it not differentiable? 6.(20 pts) Consider the Legendre equation (q) (1 - x 2 )y" - 2xy' + 2y = 0. a) Find two linearly independent solutions about x = 0. (You must solve completely any relevant recurrence relations.) b) Compute the radius of convergence for each fundamental solution in part a). c) Show that x = 1 is regular singular point of (q). d) Assuming that y = (x-1)5 I:� an (x-lt (withs E JR) is a solution near x = l, solve the indicial equation satisfied by s. Find one fundamental solution of (q) near x = l.7. Solve the following system: { x� (t) = -llx1 (t) + 6x2(t) x;(t) = -16x1(t) + 9x2(t) with initial conditions x1 (0) = -2, x2(0) = -2.

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Differential Equations
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