IZTECH Math255 Notes p5

By E. Z.

Cauchy Euler Equations

Second order linear equaitons in the form:
\(\ x^2 y +α x y + β y = 0, α,β∈ℝ,x∈ℝ^+ \)

Strategy:

By putting \(\ y=x^r \) into the DE: Chracteristic equation becomes \(\ r^2+(α - 1) r + β=0 \)

\(\ r^2+(α - 1) r + β=0 \)

Real and distinct roots

\[\ y(x)=c_1x^{r_1}+c_2x^{r_2} \]
\(\ r^2+(α - 1) r + β=0 \)

Real repeated roots

\[\ y(x)=c_1x^{r_0}+c_2ln(x)x^{r_0} \]
\(\ r^2+(α - 1) r + β=0 \)

Complex conjugate roots \(\ \small λ±iμ=r_{1,2} \)

\[\ y(x)=x^{λ}(c_1cos(μln(x))+c_2sin(μln(x))) \]

Higher Order DE's

nth order lineari nonhomogenous DE: \[\ y^{(n)}+p_1(x)y^{(n-1)}+...+p_{n-1}(x)y'+p_n(x)y=q(x) \]with initial conditions : \[\ y(t_0)=y_0 \]\[\ y'(t_0)=y_1 \]...\[\ y^{n-1}(t_0)=y_{n-1} \]
If the functions \(\ p_1, p_2, ... ,p_n,q(x)\) are continous on the open interval \(\ 𝕀 \) then there exists a unique solution \(\ y=φ(x) \) for DE.
\[\ 𝒲(y_1,y_2,...,y_{n})=det \begin{bmatrix} y_1 & y_2 & ... & y_n \\ y'_1 & y_2' & ... & y_n' \\ ... & ... & ... & ... \\ y^{n-1}_1 & y^{n-1}_2 & ... & y^{n-1}_n \end{bmatrix} \]
Functions \(\ p_1,p_2,...,p_n \) are continous on the opne interval \(\ 𝕀 \), if the functions \(\ y_1,y_2,...,y_n \) are solutions to n'th order, linear, nonhomogenous DE: \(\ ∃ t_0 ∈ 𝕀 ; 𝒲(y_1,y_2,...,y_n)(t)≠0 ⇒ \text{ \small every } y_k \text{ \small that satisfies the DE in linear combinations of them} \)
\(\ 𝒲(y_1,y_2,...,y_n)(t)≠0, m∈[1,n] \) that satisfies the DE then the set {\(\ y_1,y_2,...,y_n \)} is called fundamental set of solutions.

Homogenous Equation with constant coefficients

\(\ y^{(4)}+y=0 ⇒ r^4+1=0 ⇒ r_{1,2,3,4}=±\frac{1}{\sqrt{2}}±\frac{1}{\sqrt{2}}i\) ⇒ \(\ r_1 = e^{i π 3/4}, r_2 = e^{i π /4}, r_3 = e^{i π 7/4}, r_4 = e^{i π 5/4} \) by using these it is known how to find \(\ y_1, y_2,y_3,y_4 \)

System of First Order Linear Equations

Predator-Prey Model:
\(\ \frac{dx}{dt}=-αx+βxy \) , \(\ \frac{dy}{dt}=δy-γxy \) , \(\ α , β, δ, γ ∈ ℝ \)
\(\ x(t) \) is population of predator at time \(\ t \) , \(\ y(t) \) is population of prey at time \(\ t \)
Transform higher order DE to System of first order linear differantial equations
mass-spring oscillation: \(\ x''=- \frac{b}{m}x'- \frac{k}{m}x \)
\(\ v=x', v'=- \frac{b}{m}x'- \frac{k}{m}x \)

\(\ \begin{pmatrix} x \\ v \end{pmatrix}' = \begin{pmatrix} x' \\ v' \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ -\frac{k}{m} & -\frac{b}{m} \end{pmatrix} \begin{pmatrix} x \\ v \end{pmatrix} \)
\(\ \vec{u'} = A \vec{u}+G \)

By investigating A long time behavior of solutions even without knowing the solutions is possible.

Review on Matrices and System of Linear Equations

\(\ y(x)= \vec{v}e^{λx} \)
\(\ y'=Ay ⇒ Av=λv \)
for \(\ \vec{v} = \begin{pmatrix} v_1 \\ v_2 \end{pmatrix} \) : \(\ det(A-λI)=\begin{vmatrix} a_{11}-λ & a_{12} \\ a_{21} & a_{22}-λ \end{vmatrix}=0 \)

Real and Distinct Eigenvalues

\(\ y=c_1 v_1e^{λ_1 x}+c_2 v_2 e^{λ_2 x} \)

Real and Repeated Eigenvalues

If there is two linearly independent eigenvectors(then eigenvalues called to be complete eigenvalues):

\(\ y=c_1 v_1e^{λ x}+c_2 v_2 e^{λ x} \)

Real and Repeated Eigenvalues

If there is two linearly dependent eigenvectors:

Cayley–Hamilton Theorem:
There exists such w ≠ 0 that \(\ (A-λI)^2 w=0 \)
\(\ y=c_1ve^{λx}+c_2we^{λx} \) Even that this is a family of solutions IT IS NOT THE GENERAL SOLUTION since it does not represents all of the possible solutions.
General solution is: \(\ y=c_1ve^{λx}+c_2(w+xv)e^{λx} \)

Fundemental Matrices

Iff \(\ y_1,y_2,...,y_n \) are linearly independent solutions for an n dimentional linear system of first order homogenous equations, the set {\(\ y_1,y_2,...,y_n \)} is called to be fundemental solutions, \(\ φ \begin{bmatrix} | & | & & | \\ y_1 & y_2 & ... & y_n \\ | & | & & | \end{bmatrix} \) is called to be fundamental matrix.