Math 33B Midterm Review

Linear ODE

100

Find the general solution:
y' + y = 2

y(t) = 2 + Ce^-t

200

Find the general solution:

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

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

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:

(1+x)y' + y = cos(x)

y(x) = (sin(x) + C)/(1+x)

500

Find the general solution:

y' = cos(x) - y sec(x) 

y(x) = (x - cos(x) + C)/(sec x + tan x)

Separable ODE

100

Find the general solution:

y' = xy

y(x) = Ae^(x^2/2)

200

Find the general solution:

y' = e^(x - y)

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

300

Find the general solution:

y' = xy + y

y(x) = De^(1/2 * x^2 + x)

400

Find the general solution:

y' = x/(y + 2)

y(x) = +/- sqrt(x^2 + E)

500

Find the general solution:

x^2y' = ylnly - y'

y(x) = e^(De^(tan^-1(x)))

Exact Equations

100

Find the general solution:

(2x+y)dx + (x-6y)dy = 0

F(x,y) = x^2 + xy - 3y^2 = C

200

Find the general solution:

dy/dx = (3x^2 + y)/(3y^2 - x)

F(x,y) = x^3 + xy - y^3 = C

300

Find the general solution:

(u + v)du + (u - v)dv = 0

F(u,v) = (u^2 + 2vu - v^2)/2 = C

400

Find the general solution:

(x^2y^2 - 1)y dx + (1+x^2y^2)x dy = 0, integrating factor: u(x,y) = 1/(xy)

F(x,y) = (x^2y^2)/2 - ln x + ln y = C

500

Find the general solution:

(x^2 + y^2 - x) dx - y dy = 0, integrating factor u(x,y) = 1/(x^2 + y^2)

F(x,y) = x - 1/2 ln(x^2 + y^2)

Autonomous ODE

100

Determine the fixed points:

y' = 6 + y - y^2

y = 3, y = -2

200

Determine the fixed points:

y' = cos(2y)

y = ( (2n+1)*pi/(4)), for all natural numbers n

300

Determine the stability of fixed points:

y' = (y+1)(y - 4)

y = -1 stable, y = 4 unstable

400

Determine the stability of fixed points:

y' = 9y - y^3

y = 3,-3 stable

y = 0 unstable

500

Determine the stability of fixed points:

y' = sin(y)

y = n*pi (odd), stable

y = n*pi (even), unstable

Second Order Linear ODE

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)