M427K-Test 2-Practice

# M427K-Test 2-Practice - x = 2 2 y 00 x 1 y 3 y = 0 y(2 =-3...

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M427K—Test 2 Practice Unique #55335 Name: USE PROPER NOTATION AND SHOW ALL WORK. POINTS WILL BE AWARDED BASED ON YOUR USE OF CALCULUS. 1. A mass weighing 16lb stretches a spring 3in. The mass is attached to a viscous damper with a damping constant of 2lb · s/ft. If the mass is set in motion from its equilibrium position with a downward velocity of 3in/s, ﬁnd its position u at any time t , and determine when the mass ﬁrst returns to its equilibrium position. 1

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2. Find the general solution y = c 1 y 1 ( t ) + c 2 y 2 ( t ) + c 3 y 3 ( t ) of the following diﬀerential equation y 000 - 5 y 00 + 3 y 0 + y = 0 . Justify that y 1 ( t ) ,y 2 ( t ) ,y 3 ( t ) form a fundamental set of solutions by showing that their Wronskian is nonzero. 2
3. Find a particular solution of the following diﬀerential equation y (4) - y 000 - y 00 + y 0 = t 2 + 4 . 3

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4. Seek a power series solution of the following diﬀerential equation about the given point

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Unformatted text preview: x = 2. 2 y 00 + ( x + 1) y + 3 y = 0 , y (2) =-3 , y (2) = 1 . 4 5. Determine a lower bound for the radius of convergence of series solutions about each given point x for the following diﬀerential equation. ( x 2-2 x-3) y 00 + xy + 4 y = 0; x = 4 , x =-4 , x = 0 . 5 6. Consider the Euler equation x 2 y 00 + αxy + βy = 0. Find conditions on α and β so that: (a) All solutions approach zero as x → 0. (b) All solutions approach zero as x → ∞ . 6 7. Find all singular points of the following diﬀerential equation and determine whether each one is regular or irregular. ( x sin x ) y 00 + 3 y + xy = 0 . 7 8. Seek a power series solution of the following diﬀerential equation about the given point x = 0. 2 xy 00 + y + xy = 0 . 8...
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M427K-Test 2-Practice - x = 2 2 y 00 x 1 y 3 y = 0 y(2 =-3...

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