# MAT 204 : Differential Equations w/ Linear Algebra

ODEOne Independent Variable:
F(x,y,y',y'',...,y⁽ⁿ⁾)=0
1st Order Linear Homogeneous:
$\frac{\mathrm{du}}{}$ = k(t)u
$\frac{1}{}\frac{\mathrm{du}}{}$ = k(t)
ln|u| = ∫k(t)dt + C
u(t) = ±eC e∫k(t)dt = ce∫k(t)dt

Wronskian:
u,v are independent ← $\left|\begin{array}{cc}u& v\\ u\text{'}& v\text{'}\end{array}\right|$ ≠ 0

2nd Order Linear Homogeneous:
$\frac{\mathrm{d²u}}{}$ + k$\frac{\mathrm{du}}{}$ + pu = 0
u(t) = emt
m²emt + kmemt + pemt = 0
m² + km + p = 0
Real/Complexm'su(t)
Realm1 ≠ m2c₁em₁t+c₂em₂t
Realm1 = m2c₁emt+c₂temt
Complexm = a±bic₁eatcos(bt)+c₂eatsin(bt)

Mass-Spring-Damper:
$\frac{\mathrm{d²u}}{}$ + b$\frac{\mathrm{du}}{}$ + ω²u = 0
Situationb/2u(t)
Undamped0c₁cos(ωt) + c₂sin(ωt)
Underdamped(0,ω)e-bt/2c₁cos(ωt) + c₂sin(ωt)
Critically Dampedωe-bt/2(c₁ + c₂t)
Over Damped(ω,∞)c₁em₁t+c₂em₂t
Cauchy-Euler Equation:
$\frac{\mathrm{d²u}}{}$ + kt$\frac{\mathrm{du}}{}$ + pu = 0
u(t) = tm
t²m(m-1)tm-2 + ktmtm-1 + ptm = 0
m(m-1) + km + p = 0
Real/Complexm'su(t)
Realm1 ≠ m2c₁tm₁+c₂tm₂
Realm1 = m2c₁tm+c₂(ln t)tm
Complexm = a±bic₁tatcos(b ln t)+c₂tatsin(b ln t)

PDEMany Independent Variables:
F(x₁,...,xn , y,y',y'',...,y⁽ⁿ⁾)=0

Exact Differential Equations:

I(x,y)dx + J(x,y)dy = 0

Assume there's F Such That

1. $\frac{\mathrm{\partial f}}{}$ = I
2. $\frac{\mathrm{\partial f}}{}$ = J

Solution exists iff $\frac{\mathrm{\partial I}}{}$ = $\frac{\mathrm{\partial J}}{}$

Solve (∂/∂y) ∫ I dx = J , for F(x,y)

This gives you F(x,y)

Solution is : F(x,y) = C