Differential Equations

Problem 1

Solve the initial value problem y'+2 x y=0, y(0)=2

Problem 2

Solve the initial value problem y'=2t (y^2)(y-1), y(1)=1/2.

Problem 3

A corpse is found at 10 p.m. and is found to be 75 degrees. An hour later, the body is 73 degrees. If the ambient temperature is 70 degrees, what was the time of death? (this requires a calculator)

Problem 4

A 50 liter tank is full of a salt solution with a concentration of 3g/L.  At time 0 a new salt solution of 7g/L flows into the tank at a rate of 3 L/min, mixes thoroughly, and the mixed solution leaves the tank at the same 3 L/min rate.  Find a formula for the amount of salt in the tank at any time t≥0.

Problem 5

Find and classify the equilibrium points for the differential equation y'=sin(y)(y-5) which are between y=1 and y=10.

Problem 6

Find the general solution to the differential equation y'''-4y''+5y'-2y=0.

Problem 7

Solve the initial value problem y''+2y'+2y=0, y(0)=2, y'(0)=3.

Problem 8

Find the general solution to y''+y=4(t^2)-4t+1.

Problem 9

Find a linear differential equation with constant coefficients for which (t^2)(e^t) and (e^(2t))cos(t) are solutions.

Problem 10

Solve the initial problem y''-y=0 twice - once with initial conditions y(0)=1, y'(0)=0 and again with y(0)=0, y'(0)=1.  What are these solutions?

Problem 11

If y is a solution to y'=5y(1-(y/7)), y(0)=2, what is the limit of y as t->∞?

Differential Equations Concepts

Separable differential equations

Specific applications of first-order differential equations (population growth, mixing problems, Newton’s Law of Cooling, etc.)

Finding and classifying equilibrium points of autonomous differential equations

Linear differential equations with constant coefficients (homogenous and non-homogeneous)