We consider the differential equations /(t) = cos(t) /(t) = cos(y(t)) (1) (2) 1. We first consider ODE (1). Determine on a graph the regions of the (z, y)-plane where the slope of the tangent field is zero, positive, negative. For this same equation, draw on this same graph the typical behaviour of some particular solutions for various initial conditions. 2. Now we consider ODE (2). Determine on a graph the regions of the (z, y)-plane where the slope of the tangent field is zero, positive, negative. For this same equation, draw on this same graph the typical behaviour of some particular solutions for various initial conditions. 4. Determine graphically the possible limits of the solutions when t tends to +oo. 5. How could you prove that the solutions really tend to theses limits?
We consider the differential equations /(t) = cos(t) /(t) = cos(y(t)) (1) (2) 1. We first consider ODE (1). Determine on a graph the regions of the (z, y)-plane where the slope of the tangent field is zero, positive, negative. For this same equation, draw on this same graph the typical behaviour of some particular solutions for various initial conditions. 2. Now we consider ODE (2). Determine on a graph the regions of the (z, y)-plane where the slope of the tangent field is zero, positive, negative. For this same equation, draw on this same graph the typical behaviour of some particular solutions for various initial conditions. 4. Determine graphically the possible limits of the solutions when t tends to +oo. 5. How could you prove that the solutions really tend to theses limits?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Exercise 1
We consider the differential equations
/(t) = cos(t)
y (t) = cos(y(t))
(1)
(2)
1. We first consider ODE (1). Determine on a graph the regions of the (x, y)-plane where the
slope of the tangent field is zero, positive, negative.
For this same equation, draw on this same graph the typical behaviour of some particular
solutions for various initial conditions.
2. Now we consider ODE (2). Determine on a graph the regions of the (r, y)-plane where the
slope of the tangent field is zero, positive, negative.
For this same equation, draw on this same graph the typical behaviour of some particular
solutions for various initial conditions.
4. Determine graphically the possible limits of the solutions when t tends to +o.
5. How could you prove that the solutions really tend to theses limits?
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