1st order ODE

Question 1

N(y) * dy/dx = M(x)

Answer 1

This is a separable equation

Question 2

P(t)y’’ + Q(t)y’ + R(t)y = S(t)

Answer 2

This is a linear equation

Question 3

dy/dt = f(y)

Answer 3

This is an autonomous equation.

Question 4

M(x,y) + N(x,y)*dy/dx = 0

Answer 4

This is an exact equation

Question 5

How do you solve for a separable equation

N(y) * dy/dx = M(x)

Answer 5

To solve, ‘separate’ the dy/dx and integrate both sides

Question 6

How do you solve for a linear equation?

P(t)y’’ + Q(t)y’ + R(t)y = S(t)

Answer 6

Use the integrating factor method (multiply both sides of the equation by μ)

Question 7

How do you solve for an autonomous equation?

dy/dt = f(y)

Answer 7

An autonomous differential equation is always separable, and can be solved as such

Question 8

How do you solve an exact equation? (Not the long method)

M(x,y) + N(x,y)*dy/dx = 0

Answer 8

Check to see if M_y = N_x.
If true, you can say that phi_x = M, phi_y = N, and phi = some constant (c)
Once you’ve established this, you can essentially solve for phi as if it were a potential function

Question 9

How do you solve an exact equation? (The long method)

M(x,y) + N(x,y)*dy/dx = 0

Answer 9

Check to see if M_y = N_x.
if false, you need to see if (M_y - N_x)/N only depends on x
if THIS is true, then you can a) set μ = e^(∫(M_y - N_x)/N dx) b) multiply everything by this μ
And then you can say that phi_x = Mhat, phi_y = Nhat, and phi = some constant (c), where the hat represents the variable multiplied by this mu
Once you’ve established this, you can essentially solve for phi as if it were a potential function

Question 10

Equilibrium solutions (For autonomous equations)

Answer 10

They are constant solutions that correspond to no change or variation in the value of y as t increases. They’re also known as critical numbers, and can be found by locating the roots of f(y) = 0

Question 11

Phase Line (for autonomous equations)

Answer 11

indicates where y is increasing/decreasing via arrows on the y-axis of a graph plotting y vs. t

Question 12

asymptotically stable (for autonomous equations)

Answer 12

Whenever a graph of y vs. t converges on an equilibrium point

Question 13

asymptotically unstable (for autonomous equations)

Answer 13

Whenever a graph of y vs t diverges away from an equilibrium point

Question 14

asymptotically semi-stable (for autonomous equations)

Answer 14

whenever a graph of y vs. t converges on one side, and diverges on another