Diff EQ

growth/decay

Q = Qo*e^kt

Newton's law of cooling

(dT/dt) = k(M-T)

tank problem

x'=(r_in)(c_in)-(r_out)(c_out)

competition model

(dR/dt) = R(ar-brR-crS)

isoclines, equilibrium, stability, concavity

directly integrate

seperation of variables

(t)dt=(y)dy

Linear + homogenous?

superposition principle

Linear nonhomogenous

Find homogeneous and particular solutions

integrating factor method

dy/dt + p(t)y = f(t)
u = e^∫p(t)dt

does it exist? (check for domain issues)

is it unique? (check derivative for domain issues)

LaPlace Transform

solve for Y(s), use tables, take inverse etc.

homogeneous (RHS)

ar^2+br+c=0

real: y(h) = c1e^(r1t)+c2e^(r2t)

imaginary: y(t) = e^(at)(c1cosbt+c2sinbt)

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

particular (LHS)

form based on function: Ae^t, At^2+Bt+C, Acosbt+Asinbt
take derivative twice, plug back into RHS to solve for particular

undetermined coefficients

variation of parameters

Subtopic

eigen values

[a-lamda b
c d-lamda]

tTake determinant, solve for lamda, plug in eigenvalue for general solution. If IVP plus in those values and solve for constants