Differential Equations

Intro to Diff Eq

100

The order and linearity (linear or nonlinear) of x³(y''') + x(y') -5y = e^x 

What is third order, linear?

200

Given x = c₁cos4t + c₂sin4t, find a solution of the initial-value problem: x'' + 16x = 0, x(pi/2) = -2, x'(pi/2) = 1

What is x = -2cos4t + 1/4sin4t?

300

The model equation for population growth, where P(t) is total population at time t and k is a constant.

What is dp/dt = kP?

400

The definition of a dynamical system.

What is a system consisting of a set of time-dependent variables (state variables) with a rule that enables us to determine the state of the system? (variables + rule)

500

The diff eq model equation for the spread of disease, in terms of x (# people with disease), y (# people not exposed), k (constant), and n (fixed population)

What is dx/dt = kx(n + 1 - x) ?

First Order Diff Eq

100

The form of a separable equation.

What is dy/dx = g(x)h(y) ?

200

Solve dy/dx = y² - 4. Note: the answer is not a constant.

What is y = 2 (1+ce^4x)/(1-ce^4x) ?

300

Solve the linear equation: xdy/dx - 4y = x⁶e^x. 

What is y = x⁵e^x - x⁴e^x + cx⁴ ?

400

Solve the exact DE: (e^2y - ycosxy)dx + (2xe^2y - xcosxy + 2y)dy = 0.

What is xe^2y - sinxy + y² + c = 0 ?

500

Solve dy/dx = (-2x + y)² - 7, y(0) = 0.

What is y = 2x + 3(1 - e^6x)/(1 + e^6x) ?

Modeling with First Order Diff Eq

100

Definition of a logistic curve.

What is "graph of a logistic function"?

200

A bone is found to contain 0.1% of its original amount of C-14. Determine the age of the fossil.

What is 57,100 years ?

300

When a cake is removed from an oven, its temperature is measured at 300 degree F. Three minutes later its temp is 200 degree F. How long will it take for the cake to cool off to a room temperature of 70 degree F?

What is half an hour?

400

Solve: A student carrying a virus returns to a campus of 1000 students. If the rate at which the virus spreads is proportional to x infected students and number of uninfected students, determine the number of infected students after 6 days if after 4 days x(4) = 50.

276 students

500

The differential equation known as the density-dependent hypothesis.

What is dP/dt = Pf(P) ?

Higher Order Diff Eq (part 1)

100

A difference between a BVP and an IVP.

Any correct answer can work.

200

Solve y''' + 3y'' - 4y = 0.

What is y = c₁e^x + c₂e^-2x + c₃xe^-2x ?

300

The superposition principle.

What is "given y₁, y₂, ... y_k" are solutions of the homogeneous nth-order differential equations on an interval I, then the linear combination y = c₁y₁ + c₂y₂+ ... + c_ky_k is also a solution on the interval.

400

Solve the IVP: 3y''' + 5y'' - y' + 7y = 0.

What is y = 0?

500

Given that y₁=e^x is a solution of y'' - y = 0 on the interval from negative infinity to positive infinity, use reduction of order to find a second solution y_2.

y_{2 =} y₁integral((e^{-integral(P(x))dx})/y₁²)dx

Higher Order Diff Eq (part 2)

100

What is the assumed particular solution for a nonhomogeneous equation equalling 6e^-5x ?

What is y_p = Ae^-5x ?

200

Solve y'' + 4y' - 2y = 2x² - 3x + 6. Give y_p only.

What is y_p = -x² - 5/2x - 9 ?

300

Solve y''' + y'' = e^xcosx. Give y_p only.

What is y_p = -1/10 e^xcosx + 1/5 e^xsinx ?

400

Determine the form of a particular solution for (annihilator approach): y''' - 4y'' + 4y' = 5x² - 6x + 4x²e^2x + 3e^5x.

What is y_p = Ax + Bx² + Cx³ + Ex²e^2x + Fx³e^2x + Gx⁴e^2x + He^5x ?

500

The main contributor to the variation of parameters method.

What is Joseph Louis Lagrange?