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. Verify that the function is a particular solution of the given differential equation. Give an interval of definition / for each solution. (a) %" +zy' +y = 0; y = sin(In(z)) (b) %" + zy' +y = sec(In(z)); y = cos(In(x)) In(cos(In(z))) + (In(z)) sin(In(z)) . Draw the phase portrait and direction field for the given autonomous first-order differential equa- tions. Classify the equilibrium points as stable, unstable, or semi-stable. Then sketch some solution curves. (a) =y* 6y + 9? (b) =yly2)*(1y) . Classify the given differential equations as separable, exact, linear, homogeneous, or Bernoulli (may be more than one kind) and solve it. (a) (y* +1)dz = ycos*(z)dy (b) (3y + 2z)dx = (4y* + 6zy)dy (c) t% 4z =1 In(t) dy 23 322 2 5 e D e & () dz+ 2z +y+1)dy=0 (f) (2zy)dy (22> y*)dz =0 . lodine-131 is a radioactive liquid used in the treatment of cancer of the thyroid. After one day in storage, analysis shows that initial amount of iodine-131 in a sample has decreased by 8.3%. (a) Find the amount of iodine-131 remaining in the sample after 8 days. (b) What is the significance of your answer in part (a)? . A large tank is partially filled with 100 gallons of fluid in which 10 pounds of salt is dissolved. Brine containing % pound of salt per gallon is pumped into the tank at a rate of 6 gal/min. The well-mixed solution is then pumped out at a slower rate of 4 gal/min. Find the number of pounds of salt in the tank after 30 minutes