Kategoriak: All - solutions - stability

arabera Christopher Voss 11 years ago

326

Diff EQ

The discussion focuses on various aspects of differential equations, including their existence, uniqueness, and methods for solving them. It covers both first-order and higher-order differential equations, examining linear homogeneous and nonhomogeneous types, and explores techniques such as the LaPlace Transform, separation of variables, and integrating factors.

Diff EQ

Diff EQ

Systems of diff EQs

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

2nd order

Subtopic
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

variation of parameters

undetermined coefficients

homogeneous (RHS)
ar^2+br+c=0

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

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

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

Nth order

LaPlace Transform
solve for Y(s), use tables, take inverse etc.
does it exist? (check for domain issues)
is it unique? (check derivative for domain issues)

1st order

integrating factor method
dy/dt + p(t)y = f(t) u = e^∫p(t)dt
Linear nonhomogenous
Find homogeneous and particular solutions
Linear + homogenous?
superposition principle
seperation of variables
(t)dt=(y)dy
directly integrate
isoclines, equilibrium, stability, concavity

application based (model)

competition model
(dR/dt) = R(ar-brR-crS)
tank problem
x'=(r_in)(c_in)-(r_out)(c_out)
Newton's law of cooling
(dT/dt) = k(M-T)
growth/decay
Q = Qo*e^kt