5. Fun Fact : The chain rule can be used to find the derivative of the function

Question

5. Fun Fact : The chain rule can be used to find the derivative of the function y = |x|. Recall that for all x, (x^2) = |x|. For example, ((1)^2) = 1 = | 1|. So, (d/dx)*(|x|) = (d/dx)*( (x^2)) = (1/2)*(x^2)^(1/2)*2x = x / (x^2) = x / |x| .

(a) Show that y1 = x and y2 = |x| are both solutions to the IVP y' = x/y, y(3) = 3.

(b) Give an example of a rectangular region in the xy-plane containing the point (3, 3) such that x/y and /y*(x/y) are both continuous in the entire rectangular region. Explain your answer.

(c) Why do the two different solutions y1 = |x| and y2 = x to the IVP not contradict the existence and uniqueness theorem?

Explanation / Answer

a0for y=-x

dy/dx=-1 then,x/y=-x

therefore y=-x

for y=|x|

dy/dx=x/|x|

both satisfy the initial condition

therefore y=|x| and y=-x are solutions.

c)

here f(x,y)=x/y is not continious and also partial derivative w.r.t y also not continious.

so not satisfying the hypothesis of existance and uniquness theorem.

so do not contradict.

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