Suppose x₁ and x2 are the roots of the quadratic equation x²+bx+c = 0. (a) Use implicit differentiation to compute dx/db. (b) We know that the roots are 2= 1/(-b² X1,2 = (-b± √b²-4c). Differentiate this expression with respect to b and show that the answer is equivalent to the one you obtained in (a). (c) Find values of b and c for which the roots are very sensitive to small changes in b, and values for which they are not very sensitive.

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N.png ... If gcd(a, m) = 1, then the unique solution xo to ax = b (mod m) is equal to (1) xo = ab (mod m) (2) xo = ab (mod m), where b is the inverse of b. (3) xo = ām (mod m), where ā is the inverse of a. (4) xo = āb (mod m), where ā is the inverse of a. A Choice (2) (B) Choice (1) Choice (4) (D) Choice (3)

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I'm having trouble understanding what exactly makes a linear ordinary differential equation "linear". I know that the major advantage of the linear ODEs is that a linear combination of particular solutions gives another particular solution. Characterized by additivity and homogeneity, this means that the output for a sum of inputs is equal to the sum of outputs for each individual input (e.g., f(3) + f(5) = f(3+5)= f(8)) and scaling the input by a factor scales the output by the same factor (e.g., f(6x) =6f(x)...but how would this look? Like say if I had 4e^5(t-1) satisfying the differential equation dy/dt=5y where y(1)=4, what would that "linear combination" look like?

Please answer D,E, and F

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