Practice exam for First Midterm

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This is a practice exam for the first midterm. Feel free to drop any comment or question below. I will try to answer here as many as possible, until the day before the exam. This is a good opportunity to compare notes and work with other students. Enjoy!

  1. Prove that the function y=Ce^x is a solution of the equation y'-y=0 and find the particular solution satisfying y(0)=-1.
  2. Apply both Euler and Improved Euler methods to solving numerically the differential equation y'=y-x-1 with initial condition y(0)=1 in the interval [0, 0.5]. Use a time-step h=0.1. Prepare a table showing four-decimal-place values of the approximate solution and the actual solution at the points x=0.1, 0.2, 0.3, 0.4, 0.5.
  3. Find the general solution of the equation 3e^x \tan y + (2-e^x)\sec^2 y\, \frac{dy}{dx} =0.
  4. Find the particular solution of the equation (1+e^x)yy'=e^x that satisfies the initial condition y(0)=1.
  5. Find the equation of a curve that goes through the point (0, -2) and satisfies that the slope at any of its points is equal to three plus the y-coordinate at that point.
  6. Find the general solution of the equation xy' = \sqrt{x^2-y^2} + y.
  7. Find the general solution of the equation (x+y-2) + (x-y+4) y' =0.
  8. Find the general solution of the equation (x+y+1) + (2x+2y-1) y' = 0.
  9. Find the general solution of the equation y'+2xy=2xe^{-x^2}.
  10. Find the general solution of the equation xy'+y=y^2\ln x.
  11. Find the general solution of the second-order differential equation y''=0.
  12. Find the particular solution to the equation y''=2y^3 that satisfies the initial conditions y(0)=1 and y'(0)=1.
  13. And let’s finish with a nice punch-line: Find the general solution of the equation y''+(y')^2=2e^{-y}. If you are able to get this question in less than 30 minutes without the help of a computer, and explain to someone else step-by-step how to do it, I consider that you have mastered the material of the first midterm.
  1. Anonymous
    September 13, 2012 at 6:46 pm
    Reply

    how can you find the actual solutions to problem 2?

    • Francisco Blanco-Silva
      September 13, 2012 at 7:10 pm
      Reply

      You will have to solve the differential equation (use “jazz” for this one), get the particular solution that satisfies y(0)=1, and plug the values of x=0.1, 0.2, 0.3, 0.4, 0.5

  2. Anonymous
    September 19, 2012 at 8:53 pm
    Reply

    for number 12. I’ve gotten to dy/dx = sqrt(y^4 +c) assuming that is correct, I’m not sure where to go from there, it seemed separable but was having some trouble with that.

  3. Anonymous
    September 19, 2012 at 11:28 pm
    Reply

    For number 2 is the solution y=2+x-e^x if yes, why do I get answers that are way different than the ones I got from using Eulers method?

    • Anonymous
      September 20, 2012 at 5:42 pm
      Reply

      If you got similar answers for both EM’s then that means your solution is wrong.

  4. Anonymous
    September 20, 2012 at 5:52 pm
    Reply

    is the answer to number 5 : y’=(y-1)/x?

  5. Anonymous
    September 20, 2012 at 8:25 pm
    Reply

    for number 5 i got y=e^x – 3

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