5.1 DE Basics and Exact ODEs
DE Basics
Linear Nth Order ODE:
(d^n)(y)/(d(x)n)+(p_1)(x)(d^n-1)(y)/(d(x)(n-1))+⋯+(p_n-1)(x)d(y)/d(x)+(p_n)(x)*y(x)=g(x)
If g(x)=0 then it is homogeneous.
If g(x)≠0 then it is non-homogeneous.
(y_gen)*(x)=(y_hom)*(x)+(y_par)*(x)
If the Wronskian is zero then the solutions aren't linearly independent:
W((y_1),⋯,(y_n))=|[(y_1),(y_2),⋯,(y_n)],[(y_1^′),(y_2^′),⋯,(y_n^′)],[,,,],[(y_1)(n-1),(y_2)(n-1),⋯,(y_n)(n-1)]|
If k=λ+i*μ then:
(y_hom)=e(λ*x)*((c_1)*cos(μ)*x+(c_2)*sin(μ)*x)
The Chain Rule:
∂(u)/∂(t)=∂(u)/∂(x)∂(x)/∂(t)+∂(u)/∂(y)∂(y)/∂(t)
Exact ODEs
M(x,y)+N(x,y)d(y)/d(x)=0
⇔
(ψ_x)=∂(M)/∂(y)=∂(N)/∂(x)=(ψ_x)
ODE Methods & Laplace Transforms -> click
D'Alembert: y(x)=old solution×u(x).
Variation of parameters: (y_par)=(∑_j=1^n)((u_j)(x)*(y_j)(x))
Diagonalization: x=T*y:
T*(y^′)=A*T*y+g(t)⇒d(y)/d(t)=D*y+T(-1)*g=D*y+h
Systems, Stability, Fourier, PDEs -> click
First solution:
x(1)=e(λ*t)*(ξ_λ)
Second solution:
x(2)=t*ℇ(λ*t)*(ξ_)+ℇ(λ*t)*η
where (ξ_)=(A-λ*𝕀)*η. Third solution:
x(3)=(t2)/2*e(λ*t)*(ξ_λ)+t*ℇ(λ*t)*η+e(λ*t)*ζ
where η=(A-λ*𝕀)*ζ.
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