Test 2 Review For Exam PDF

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Math 320

New York Institute of Technology Mathematics Department, CAS Review Exam 2

Fall 2020

1. Check if separable and find the general solution √ of the differential equation 2 (x − 4xy )dx + xy x + 2dy = 0 2. Solve the Linear first order differential equation to find the particular solution. dy − y = x3 e3x , satisfying y(1) = 0 x dx 3.Solve the Bernoulli differential Equation

dy dx

− 2yx = xy2

4(a) Justify whether or not the following first order differential equations are exact: Do not find solution (a) (sin(x + y) + 2xy)dx − (sin(x + y) + x2 )dy = 0 4(b) Verify that the following first order differential equation is exact and find the particular solution. (1 + ex y + xex y)dx + (xex + 2)dy = 0 y(0) = 1 5. Find the general solution to the differential equation 2 dy (a) ddt2y + 8 dt + 8y = 0 (b) Find the general solution of the given equation. d2 y − 2 dy dt + 13y = 0 dt2 6. Find a solution to the initial value problem d2 y ′ + 2 dy dt + y = 0, satisfying y(0) = 1, y (0) = −3 dt2 7.The differential equation my′′ + by′ + ky = 0 that describes the model for a vibrating spring system with damping. A damped mass-spring osscillator consisting of a mass 14 kg attached to a spring fixed at one end is displaced − 12 m to the left (negative direction) of the equilibrium position and given a velocity of −2m/s to the left (negative direction). Taking damping constant as 2kg/sec and stiffness as 8kg/sec2 , find (i)the equation of motion and sketch an accurate graph labeling maxima and minima. (ii)Analyze the behavior of displacement as t −→ ∞. (iii)Also the max displacement to left after it starts motion.

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2 8. Use the variation of parameter method to solve d2 y + 9y = sec(3t) dx2 9. (a). Using the translation property find L{e−t (t2 − 3te2t )}. (b). Find L{4t cos2 t − 3} (Use: cos(2t) = 2 cos2 t − 1) 10(a)Use the method of undetermined coefficients to find the form of yp ( DO NOT SOLVE FOR UNKNOWN) For ay′′ +by′+cy = Ctm eαt cos(βt), you may use yp (t) = ts (Am tm +...+A1 t+A0 )eαt cos(βt)+ ts (Bm tm + ... + B1 t + B0 )eαt sin(βt) with s=0 if r is not a root, s=0 if α + βi is not aroot and s=1 if α + βi is a root of the auxiliary equation. y′′ − 2y′ + 2y = 4t cos t 10(b)Use the method of undetermined coefficients to find the general solution of y′′ + 3y′ − 4y = 5tet ′′ ′ m rt (hint: For ay + by + cy = Ct e , you may use yp (t) = ts (Am tm + ... + A1 t + A0 )ert with s=0 if r is not a root, s=1 if r is a simple root and s=2 if r is a double root of the auxiliary equation.)...