## Practice exam for First Midterm

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. September 13, 2012 at 6:46 pm

how can you find the actual solutions to problem 2?

• September 13, 2012 at 7:10 pm

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. September 19, 2012 at 8:53 pm

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. September 19, 2012 at 11:28 pm

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?

• September 20, 2012 at 5:42 pm

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

4. September 20, 2012 at 5:52 pm

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

5. September 20, 2012 at 8:25 pm

for number 5 i got y=e^x – 3