Accessible Differential Equations ODE Flow Chart

Decision: Does the ODE look like this?
d y d t plus g of t times y equals b of t \[ \frac{dy}{dt} + g(t)\cdot y = b(t) \]

→ No — use Separation of Variables

Separation of Variables

The ODE should look like:

d y d t equals h of y times f of t \[ \frac{dy}{dt} = h(y)\cdot f(t) \]

Apply Separation of Variables. The solution can take many forms.

→ Yes — use Integrating Factor Method

Integrating Factor Method

Apply the Integrating Factor Method. The solution takes the form:

y of t equals 1 over mu of t, times the integral of mu of t times b of t d t \[ y(t) = \frac{1}{\mu(t)}\int \mu(t)\,b(t)\,dt \]

where the integrating factor is:

mu of t equals e to the integral of g of t d t \[ \mu(t) = e^{\int g(t)\,dt} \]

Decision: Are the eigenvalues real or complex?

→ Complex eigenvalues

Complex Eigenvalues

The eigenvalues take the form:

lambda sub 1 comma 2 equals alpha plus or minus i times beta \[ \lambda_{1,2} = \alpha \pm i\cdot\beta \]

Find one complex eigenvector (it will contain \(i\)). Separate it into real and imaginary parts:

Y vector Real of t plus i times Y vector Imaginary of t \[ \vec{Y}_\text{Real}(t) + i\cdot\vec{Y}_\text{Imag}(t) \]

Solution:

Y vector of t equals k sub 1 times Y vector Real of t, plus k sub 2 times Y vector Imaginary of t \[ \vec{Y}(t) = k_1\cdot\vec{Y}_\text{Real}(t) + k_2\cdot\vec{Y}_\text{Imag}(t) \]

Note: the \(i\) term has become \(k_2\).

→ Real eigenvalues

Real Eigenvalues — Repeated or Distinct?