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Is it separable?

Find the homogeneous solution

Find the particular solution

Solution: y(t) = c1e^(r1t) + c2e^(r2t)

Solution: y(t) = c1e^(λt)v + c2te^(λt)v + c2e^(λt)u

Can you integrate directly?

Use eigenvectors and eigenvalues

Variations of parameters method

Solution: integrate and solve for y

Solve using laplace transforms

Form the characteristic polynomial

Solution for non-homogeneous: y(t) = (homogenous solution) + (particular solution)

Repeated Roots

Solution: y(t) = c1e^(r1t) + c2te^(r2t)

Solve using eigenvectors and eigenvalues if the O.D.E. can be written as a system of first order differential equations; otherwise, use laplace transforms.

Solution: y(t) = c1e^(αt)[cos(βt)p - sin(βt)q] + c2e^(αt)[sin(βt)p - cos(βt)q]

Is it homogeneous?

Solution: separate, integrate, and solve for y

Is the O.D.E. second order?

Complex (Imaginary) Roots

Solution: y(t) = e^(αt)[c1cos(βt) + c2sin(βt)]

Subtopic

Undetermined coefficients method

Use the integrating factor method

Solution: calculate the integrating factor, multiply by the integrating factor, integrate, and solve for y.

Real Roots

Solution: y(t) = c1e^(λ1t)v1 + c2e^(λ2t)v2

Can the O.D.E. be written as a system of first order differential equations?

Solving O.D.E.'s

Is the O.D.E first order?