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1. (a) State the basic existence and uniqueness theorem for the first order initial value problem  y’=f(x,y)    y(x0)=y0

(b) . State the basic existence and uniqueness theorem for the first order linear initial value problem y’+p(x)y=g(x)   y(x0)=y0

(c) What is the best statement that you can make about the existence and uniqueness of a solution of the initial value problem y’+sin(x)y=1/x   y(1)=6

(d) What is the best statement that you can make about the existence and uniqueness of a solution of the initial value problem y’+sin(x)y³=1/x   y(1)=6

(e) What is the best statement that you can make about the existence and uniqueness of a solution of the initial value problems

exp(x)*y’+2y/(x*(x-3))=cos(x)     y(1)=5

2. An object is constrained to move horizontally and is subject to (only) a restorative force F=-kx^4, where k>0 is a constant.

(a) Find a relation between the velocity v and the position x assuming that the initial position is x0  and initial velocity v0.

(b) If the mass m is given in kg and  x is given in meters, what units for the constant k ?

3. A tank contains 1000 Liters of a solution containing100 kg of salt dissolved in water. Pure water is pumped into the tank at a rate of 5L/sec.

The mixture is pumped out at the same rate. Find the amount of salt in the tank at any time.

4. Solve the initial value problem x²y’+3xy=x^5    y(1)=9/7

5. Solve the initial value problem: y²dy/dx-x³=0    y(0)=1

6. Solve the initial value problem: 3x²exp(y)+(x³exp(y)+y)dy/dx  y(2)=2

7. For the initial value problem y’=x²y  y(0)=3

(a) Write down the values for the approximate solution obtained using Euler’s method with the points x0=0, x1=0.1 , x2=0.2 , x3=0.3 , x4=0.4 .

(b) Solve the initial value problem and compare the values of the solution with the approximate values that you found in part .What is the relative error at x4?

8. For the o.d.e. y’=y²(y-1)

Draw a rough sketch of the direction field. Find the critical points and decide if they are stable or unstable.

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