1. [10/16 Points] PREVIOUS ANSWERS In this question, you Will solve the differential equation % : e7>'tZ 7 497V. This differential equation is separable, in other words, we can write it in the form %{L : p(t) q(y). What aie the two functions p(t) and g(y)7 pm: [974) e? :10) = 97-" J Now separate the variables. Integrate both sides of the resulting equation With respect to t and replace 310% by dy in the integral on the left side. dt f [7," I," :f (,2 7 4) (If W J Do the integrations on both sides. Use k to stand for your constant of integration for the left side and K to stand for your constant of integration for the right side. [Lower case it on left side, upper case Kori right side.) Integrating left side gives Integrating right side gives: 7 7 .11 + k X We can combine the two constants of integration into a single constant, C, that appears as +6 on the right side of our equation. (We do this by subtracting k fiorn both sides of the equation ) Write the new constant C in terms of k and K c: r+(K7k) X Note: For the rest of this problem and for all problems following in this assignment, whenever we do separation of variables, we will use a single constant of integration, adding Cto the right side. Even if it is possible to simplify (like, for instance, when we wroleA : e': in an example in class), you should leave your answer in terms of C. Finally, rearrange to get y as a function of I (The constant of integration, C, will appear in youi answer) y: r+(K7k)+C Submii Answer 2. [016 Points] PREVIOUS ANSWERS Solve the differential equation. (Note: The constant of integration, C, Will appear in youi answer.) a : 2y dt [9 y : 7 %Iog(7t2 +5) 3. [016 Points] PREVIOUS ANSWERS Solve the differential equation. (Note: The constant of integration, C, will appear in youi answer.) Viewing 56 ved Work Raver? to Last Response PREVIOUS ANSWERS Solve the differential equation. There are two solutions. One solution is given. (Note: The constant of integiation, C, will appear in your answer. Hint/Tip: Since 1 + t2 is always positive, vou do not need to use absolute values after integrating.) Submit/Answer 4. [0/6 Points] (1 + IZW' : {V2 y:0 7 1 y= 10g(1+t2)y?- C 5. [016 Points] PREVIOUS ANSWERS Solve the differential equation. (Note: The constant of integration, C, will appear in youi answer.) 5 6480/4" + Krl 6f: Submii