MATH:2560 Review

Homogeneous Solutions

100

Find the General Solution of y''+4y'+3y=0

What is y(x)=C₁e^-x+C₂e^-3x

200

Find the General Solution of a differential equation that has the characteristic equation of r²(r − 2)³

What is C₁+C₂x+(C₃+C₄x+C₅x²)e^2x

300

Find the General Solution of the differential equation that has the characteristic equation of (r²+1)(r²+4)

What is C₁cos(x)+C₂sin(x)+C₃cos(2x)+C₄sin(2x)

400

Find the General Solution of the differential equation that has the characteristic equation of

(r²-4)(r²+2r+5)=0

What is C₁e^2x+C₂e^-2x+(C₃cos(2x)+C₄sin(2x))e^-x

500

Solve the IVP

y''-6y'+9y=0; y(0)=2, y'(0)=1

What is y = (2 - 5x)e^3x

Particular Solutions

100

Which of the following can we NOT use the method of undetermined coefficients for?

A) 3e^2x    B) x    C) 5sin(x)    D) tan(x)  E) xe^-xcos 2x

What is D) tan(x)

200

A forced, undamped spring-mass system satisfies 

x′′ + 4x = 5 cos 3t. What is a particular solution?

What is -cos(3t)

300

Find the Particular Solution of y''+2y'+y=4e^-x

What is 2x²e^-x

400

What is the appropriate form of a particular solution of y′′−2y′+5y = e^xcos(2x)?

(Do not solve for the constants)

What is xe^x(Acos(2x)+Bsin(2x)

500

Which is the appropriate form of a particular solution of y′′ − 2y′ + 5y = ex cos 2x?

(Do not solve for the constants)

What is y_p = xe^x(Acos 2x + Bsin 2x)

Wronskians

100

What is the formula for a 2x2 determinant?

What is ad-bc

200

Compute W(cos(3x), sin(3x))

What is W=3

300

Compute W(x, xe²)

What is x²e^x

400

Compute W(e^x, e^2x, e^3x)

What is 2e^6x

500

What does the Wronskian tell us?

Whether a set of solutions is linearly independent.

Spring/Circuits

100

A 2 kg mass hangs on a spring with spring constant 18 N/m. There is no damping and no external force. What is the period of the resulting oscillation?

What is 2pi/3 s

200

An undamped spring-mass system satisfies x′′+4x=0 with x(0) = 1 and x′(0) = 2. What is the amplitude of the motion?

What is √2 

300

A 4 kg mass stretches a spring 0.2 m at equilibrium (take g = 9.8 m/s2). With no damping, what is the natural circular frequency of the system?

What is 7 rad/s

400

Name each type of damping and the instances of roots that induce them

2 real and distinct - Overdamped

2 real and repeated - Critically Damped

Complex Conjugate - Underdamped

Purely Imaginary - Undamped

500

A series RLC circuit with L = 2 H, R = 12 Ω, C = 1/18 F and no applied voltage has charge satisfying 

2Q′′ + 12Q′ + 18Q = 0. 

Classify the motion and give the general solution for Q(t)

What is Critically damped, Q = (C₁ + C₂t)e^-3t