Chapter 2

Question 1

What form do separable first order ODEs take?

Answer 1

y’ = a(x) b(y)

WITH y(x0) = y0

Question 2

How to solve a seperable first order ODE

Answer 2

1) rearrange so the equation is in the form 
dy/dx = a(x) b(y)
2) divide by b(x)
3) multiply by dx
b(y) dy  = a(x) dx
4) integrate both sides and +c
5) rearrange for y = f(x)
6) substitute in initial conditions to find the explicit solution

Question 3

What form do equations solved by the integrating factor method take?

Answer 3

y’ + P(x)y = Q(x)

WITH y(x0) = y0

Question 4

How to solve a linear first order ODE with the integrating factor method

Answer 4

1) rearrange so that the equation is in the form
y’ + P(x)y = Q(x)
2) calculate the integrating factor
IF = e^(∫P(x)dx)
3) multiply the equation by the integrating factor
4) apply the product rule to the right hand side of the equation
5) multiply by dx
6) integrate
7) arrange for y = f(x)
8) substitute in any initial conditions to find the explicit solution

Question 5

What form do first ODEs of homogeneous degree take?

Answer 5

dy/dx = f(y/x)

Question 6

How to solve linear first order ODEs of homogeneous degree

Answer 6

1) rearrange so that the equation is in the form,
y’ = f(y/x)
2) substitute y and y’ using
y(x) = xv(x)
3) simplify the equation
4) the equation should now be in a form that can be solved using the separable or integrating factor method
5) once you have an expression for v, substitute into y = xv to find y

Question 7

What form does a Bernoulli equation take?

Answer 7

y’ + P(x)y = Q(x)y^n

where n≠0,1

Question 8

How to solve a Bernoulli equation

Answer 8

1) rearrange the equation so that it is in the form,
y’ + P(x)y = Q(x)y^n
2)use the substitution,
z(x) = y^(1-n), to find dz/dx
3) divide the equation by y^n
4) substitute in the values from (2) so you have an ODE in terms of z and x
5) this equation will be solvable by the integrating factor method, solve for z
6) using z=y^(1-n) to find y