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CAS PROJECT. Direction Fields. Discuss direction fields as follows. (a) Graph portions of the direction field of the ODE (2) (see Fig. 7), for instance, \(-5 \leqq x \leqq 2,-1 \leqq y \leqq 5\). Explain what you have gained by this enlargement of the portion of the field. (b) Using implicit differentiation, find an ODE with the general solution \(x^{2}+9 y^{2}=c(y>0)\). Graph its direction field. Does the field give the impression that the solution curves may be semi-ellipses? Can you do similar work for circles? Hyperbolas? Parabolas? Other curves? (c) Make a conjecture about the solutions of \(y^{\prime}=-x / y\) from the direction field. (d) Graph the direction field of \(y^{\prime}=-\frac{1}{2} y\) and some solutions of your choice. How do they behave? Why do they decrease for y > 0?
Chapter 1, Problem 16(choose chapter or problem)
CAS PROJECT. Direction Fields. Discuss direction fields as follows.
(a) Graph portions of the direction field of the ODE (2) (see Fig. 7), for instance, \(-5 \leqq x \leqq 2,-1 \leqq y \leqq 5\). Explain what you have gained by this enlargement of the portion of the field.
(b) Using implicit differentiation, find an ODE with the general solution \(x^{2}+9 y^{2}=c(y>0)\). Graph its direction field. Does the field give the impression that the solution curves may be semi-ellipses? Can you do similar work for circles? Hyperbolas? Parabolas? Other curves?
(c) Make a conjecture about the solutions of \(y^{\prime}=-x / y\) from the direction field.
(d) Graph the direction field of \(y^{\prime}=-\frac{1}{2} y\) and some solutions of your choice. How do they behave? Why do they decrease for y > 0?
Questions & Answers
QUESTION:
CAS PROJECT. Direction Fields. Discuss direction fields as follows.
(a) Graph portions of the direction field of the ODE (2) (see Fig. 7), for instance, \(-5 \leqq x \leqq 2,-1 \leqq y \leqq 5\). Explain what you have gained by this enlargement of the portion of the field.
(b) Using implicit differentiation, find an ODE with the general solution \(x^{2}+9 y^{2}=c(y>0)\). Graph its direction field. Does the field give the impression that the solution curves may be semi-ellipses? Can you do similar work for circles? Hyperbolas? Parabolas? Other curves?
(c) Make a conjecture about the solutions of \(y^{\prime}=-x / y\) from the direction field.
(d) Graph the direction field of \(y^{\prime}=-\frac{1}{2} y\) and some solutions of your choice. How do they behave? Why do they decrease for y > 0?
ANSWER:Step 1 of 6
Given:- The ODE equation is \(\frac{{dy}}{{dx}} = y + x\).