Transcribed TextTranscribed Text

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.

Solution PreviewSolution Preview

This material may consist of step-by-step explanations on how to solve a problem or examples of proper writing, including the use of citations, references, bibliographies, and formatting. This material is made available for the sole purpose of studying and learning - misuse is strictly forbidden.

Differential Equations
    $40.00 for this solution

    PayPal, G Pay, ApplePay, Amazon Pay, and all major credit cards accepted.

    Find A Tutor

    View available Differential Equations Tutors

    Get College Homework Help.

    Are you sure you don't want to upload any files?

    Fast tutor response requires as much info as possible.

    Upload a file
    Continue without uploading

    We couldn't find that subject.
    Please select the best match from the list below.

    We'll send you an email right away. If it's not in your inbox, check your spam folder.

    • 1
    • 2
    • 3
    Live Chats