Ch. 5 Differential Equations

Question 1

How to solve a differential equation of the form f(x,y) = X(x)Y(y)

Answer 1

Separate the function

–> ∫1/Y(y) dy = ∫X(x) dx

Question 2

How to solve a differential equation of the form
f(x,y) = -p(x)y + q(x)
or y’ + p(x)y = q(x)

Answer 2

Use an integrating factor:
I(x) = exp(∫p(x))
y = 1/I(x) ∫I(x)q(x) dx

Question 3

How to show a differential equation is exact

Answer 3

The ODE M(x,y) dx + N(x,y) dy = 0 is exact iff

∂M/∂y = ∂N/∂x

Question 4

How to solve an exact differential equation in the form M(x,y) dx + N(x,y) dy = 0

Answer 4

We have M = ∂g/∂x and N = ∂g/∂y

To find g, integrate M with respect to x, the constant of integration is φ(y)

To find φ, plug the expression for g into N:
∂g/∂y = N (differentiate expression for g with respect to y) then set equal to N

Set g = c and rearrange for y

Question 5

How to make a non exact differential equation in the form M(x,y) dx + N(x,y) dy = 0 exact

Answer 5

Multiply equation by integrating factor (will be given)

Question 6

How to solve a differential equation in the form y’ +p(x)y = q(x)y^n

Answer 6

Substitute v = y^1-n, find v’ and substitute in

Note: if v = 1/y, v’ = -y’/y^2

Question 7

Characteristic equation: distinct real roots λ1, λ2

Answer 7

y = Ae^λ1x + Be^λ2x

Question 8

Characteristic equation: repeated real roots

Answer 8

y = Ae^λx + Bxe^λx

Question 9

Characteristic equation: complex roots a +bi

Answer 9

y = e^ax(Acos(bx) + Bsin(bx))