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Jm
Numerade Educator

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Problem 7 Easy Difficulty

(a) What can you say about a solution of the equation $ y{'} = - y^{2} $ just by looking at the differential equation?

(b) Verify that all members of the family $ y = 1/(x + C) $ are solutions of the equation in part (a).

(c) Can you think of a solution of the differential equation $ y^{'} = - y^{2} $ that is not a member of the family in part (b)?

(d) Find a solution of the initial-value problem.

$ y^{'} = - y^2 $ $ y (0) = 0.5 $

Answer

(a) (a) since the derivative $y^{\prime}=-y^{2}$ is always negative (or 0 if $y=0$ ), the function $y$ must be decreasing (or equal to 0 ) on any interval on which it is defined.
(b) $$=-y^{2}=\mathrm{RHS}$$
(c) $y=0$
(d) $$C=2, \text { so } y=\frac{1}{x+2}$$

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Video Transcript

in this question, we have first derivative off. Why equal negative voice square. So if we differentiate both sides, we will have Why Double dash equal. Negative two y want supply. Why dash? So if we replace, why dash with negative or square? So we will get why Dublin dash equal two y porcine So now we have Why does equal negative y or two So the first derivative is negative than the solution off. This differential equation is decreasing, but also we have the second relative equal to wipe our three. So the second drifted have the same sign as the y value. So the graph off the solution is concave up. When the graph is above the X axis and concave down when it's below the X axis, this is the part a in the question 40 importante we have another function which is why equals one over X plus c. If we differentiate both sides, we will get why dash equals negative one over x plus c all square. So we have satisfied the following differential equation, which is why dash equal negative, wide squish import see while zero Why equals zero is also solution to why dash equal negative. Why Square? And it's not a member off the family off the function. Why equal one over X plus C or what? Early in this question, we will substitute X equals zero and why equal 0.5 toe Get the value off. See? So we have 0.5 equals 1/0 plus C, so in this case, C equals to.