Math 33B Final Review

2nd Order Linear, Homogenous

100

Find the roots of the characteristic equation associated with:

y'' - y' - 2y = 0

lambda = 2,-1

200

Find the roots of the characteristic equation associated with:

y'' - 2y' + 4y = 0

lambda = 1+\- \sqrt{3}

300

Find the general solution of the equation:

3y'' - 2y' - y = 0

y(t) = C_1 e^(-t/3) + C_2 e^t

400

Find the general solution of the equation:

4y'' + 4y' + y = 0

y(t) = (C_1 + C_2 t)e^(-t/2) 

500

Find the solution of the initial value problem: 

y'' - 2y' + 17y = 0, y(0) = -2, y'(0) = 3

y(t) = e^t (-2 cos 4t + (5/4) sin 4t)

2nd Order Linear, Inhomogeneous

100

Find a particular solution for the given ODE:

y'' + 2y' + 5y = 12e^-t

y = 3e^-t

200

Find a particular solution of the ODE:

y'' + 7y' + 6y = 3sin(2t)

y = -(21/100) cos(2t) + (3/100) sin(2t)

300

Find the solution of the ODE:

y'' - 2y' + y = t³, y(0) = 1, y'(0) = 0

y(t) = (-23 + 5t)e^t + t³ + 6t² + 18t + 24

400

Find a particular solution the ODE:

y'' - 2y' + y = e^t

y = 1/2 t² e^t

500

Find a particular solution to the ODE:

y'' + 9y = tan(3t)

y = -1/9*cos(3t)*ln|sec(3t) + tan(3t)|

Linear Algebra

100

Find the null space of 

A = [2 1; 4 2]

t[-1/2; 1]

200

Find the null space of 

A = [1 0 2 -2; 0 1 3 -1]

s(-2; -3; 1; 0) + t(2; 1; 0; 1)

300

Find a basis and dimension of the null space of

A = [3 -1 0 1; -1 1 1 0; 1 1 3 2; -3 3 3 0]

x_1 = (0; 1; -1; 1)

dim = 1

400

Find the determinant of the matrix

A = [1 0 4; -3 3 -2; 4 -1 -2]

-44

500

Find the eigenvalues of 

A = [-1 -4 -2; 0 1 1; -6 -12 2]

lambda = -1,1,2

Linear Systems

100

Convert this 2nd order ODE into a linear system:

2y'' - 3y' + y = 0

z' = [0 1; -1/2 3/2]z

200

Convert this 4th order ODE into a linear system:

y'''' + 6y'' - 12y' + 8y = 0

z' = [0 1 0 0; 0 0 1 0; 0 0 0 1; -8 12 -6 0]

300

Find the general solution to the system y' = Ay

A = [-5 1; -2 2]

y(t) = C₁e^-4t(1; -1) + C₂e^-3t(1; 2)

400

Find the general solution to the ODE y' = Ay

A = [3 -1; 1 1]

y = e^2t(C₁[1; 1] + C₂[1; 0])

500

Find the general solution to y' = Ay

A = [-1 -2; 4 3]

y(t) = C₁e^t(cos(2t); -cos(2t) + sin(2t)) + C₂e^t(sin(2t); -cos(2t) - sin(2t))

Misc. 1st Order

100

Find the general solution:

y' + (2/x)y = cos x/x^2

y(x) = (sin(x) + C)/x^2

200

Find the general solution:

y' = e^{(x - y)}

y(x) = ln(e^x + C)

300

Find the general solution:

x' - 2x/(t+1) = (t+1)²

x(t) = t(t+1)² + C(t+1)² 

400

Find the general solution:

y' = x/(y + 2)

y(x) = +/- sqrt(x² + E)

500

Find the general solution:

dy/dx = (3x² + y)/(3y² - x)

F(x,y) = x³+ xy - y³ = C