Differential-Difference Equations by Richard Bellman
By Richard Bellman
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Extra resources for Differential-Difference Equations
Example text
0 sec before falling. Neglecting air resistance, with what velocity was the ball thrown? 5. 00 sec before it hits the ground. If air resistance is neglected, answer the following questions: a. What was its final velocity just as it hit the ground? b. What was the average velocity during the fall? c. How high was the building? 6. 00 sec later you see it hit the ground. Neglecting air resistance, how high is the cliff? 7. Superman is flying at treetop level near Paris when he sees the Eiffel Tower elevator start to fall (the cable snapped).
2 Using the condition that the object fell from rest, so that v(0) = 0, we can determine the constant c and solve for v(t). We have v(t) = 4 − 4e−8t as the velocity of the object at any time. 3. 3 shows the behavior of the velocity over time. Analytically, we see that v(t) approaches 4 as t → ∞. This value is known as the limiting or terminal velocity of the object. Now since dx dt = v(t), we have dx = 4 − 4e−8t . dt This is easily integrated to obtain x(t) = 4t + 12 e−8t + c. If we take the initial position of the object as zero, so that x(0) = 0, then 1 1 x(t) = 4t + e−8t − .
1. SOME BASIC TERMINOLOGY 9 18. Find values of r for which y(x) = xerx is a solution to y + 4y + 4y = 0 on (−∞, ∞). 19. Classify the differential equations by specifying (i) the order, (ii) whether it is linear or nonlinear, and (iii) whether it is an initial-value or boundaryvalue problem (where appropriate). a. 3y + y = sin x b. y + sin y = 0 c. y (3) + (sin x)y (2) + y = x, y(0) = 1, y (0) = 0, y (0) = 2 d. y + ex y = y 4 , y(0) = 0 e. y + y − y = 0 f. y + ex y + y 2 = 0, y(0) = 1, y(π) = 0 20.