Mathematics

Hints and Answers

Section 1.2

16. The curves x^2 + c y^2 = 1 are ellipses (c>0), vertical lines (c=0), and hyperbolas (c<0), all going through (-1,0) or (1,0). (The ellipses go through both.) Why can so many solutions go through a single point?
26. No, the initial value problem is not guaranteed to have a unique solution. f(x,y) is continuous at the point in question, but the partial derivative of f with respect to y is not.

Section 1.3

14. The solutions are y=x (isocline for slope 1), y=-x (isocline for slope -1), and a lot of hyperbolas.

Section 2.4

16. x*e^(xy) + x/y^2 = C.

Section 3.4

13, 24, 25 are not linear equations.
24. v = beta * ln ( m_0 / (m_0 - alpha *t) ) - g*t.

Section 9.5

6. Eigenvectors are r_1 = r_2 = -1 and r_3 = 2 with associated eigenvectors u_1 = s col(-1,1,0), u_2 = u col(-1,0,1), and u_3 = s col(1,1,1).
16. c_1 e^(-10t) col(2,0,-1) + c_2 e^(5t) col(0,1,0) + c_3 e^(5t) col(1,0,2)

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