2) First Order ODEs

Question 1

How do you solve a non-separable, first-order, linear ODE

Answer 1

Question 2

What two parts make up the general solution of an ODE

Answer 2

• Particular solution
• Homogenous solution

Question 3

What is the existence and uniqueness theorem for linear ODEs

Answer 3

Consider the first order linear initial value problem,
y′(x) + p(x)y(x) = q(x)for all x ∈ [a, b] y(x0) = y0,
where x0 ∈ [a, b] and y0 ∈ R. If p, q : [a, b] → R are continuous functions, then there exists a global solution y : [a, b] → R satisfying (16). Furthermore, this solution is unique.

Question 4

What is a first-order ODE of homogeneous type

Answer 4

Question 5

How do you solve an ODE of homogenous type

Answer 5

Question 6

What is a phase portrait

Answer 6

A plot of y’ as a function of y

Question 7

What direction do you plot arrows on a phase portrait

Answer 7

• If N > 0 then N’ < 0. Leftward arrow along the N-axis
• If N = 0 then N’ = 0
• If N < 0 then N’ > 0. Rightward arrow along the N-axis