By separating variables, ﬁnd the solution of the following initial value problems in explicit form: 1. y′ = 2x/(y + x2 y), y(0) = −2. 2. y′ = xy3 (1 + x2 )−1/2 , y(0) = 1. 3. y′ = 3x2 /(3y2 − 4), y(1) = 0. (Leave solution as a cubic equation for y.) 4. y′ = (2 − ex )/(3 + 2y), y(0) = 0. 5. Find the solution of the following initial value problem with x = x(t) and x = dx/dt: ˙ x + 2x = te−2t , ˙ 6. Consider the initial value problem: 2 1 x + x = 1 − t, ˙ 3 2 x(0) = x0 . x(1) = 0.
Find the value of x0 for which the solution touches, but does not cross, the t-axis. 7. Find the value of x0 for which the solution of the initial value problem x − x = 1 + 3 sin t, ˙ remains ﬁnite as t → ∞. 8. Consider the initial value problem 3 x − x = 3t + 2et , ˙ 2 x(0) = x0 . x(0) = x0
Find the value of x0 that separates solutions that grow positively as t → ∞ from those that grow negatively. How does the solution that corresponds to this critical value of x0 behave as t → ∞? 9. A home buyer can aﬀord to spend no more than $800/month on mortgage payments. Suppose that the interest rate is 9% and that the term of the mortgage is 20 years. Assume that interest is compounded continously and that payments are also made continuously. (a) Determine the maximum amount that this buyer can aﬀord to borrow. (b) Determine the total interest paid during the term of the mortgage. 10. A sky diver weighing 82 kg (including equipment) falls vertically downward from an altitude of 1500 meters, and opens the parachute after 10 sec of free fall. Assume that the force of air resistance is k1 |v|, with k1 = 1.65 kg/s, when the parachute is closed and k2 |v|, with k2 = 26.4 kg/s, when the parachute is open. (a) Find the speed of the sky diver when the parachute opens. (b) Find the distance fallen before the parachute opens. (c) What is the limiting velocity vL after the parachute opens? (d) Determine how long the sky diver is in the air after the parachute opens. (e) Sketch the graph of velocity...
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