5.1 DE Basics and Exact ODEs

DE Basics

Linear Nth Order ODE:

$\frac{d^n\left(y\right)}{{d\left(x\right)}^n}+p_1\left(x\right)\frac{d^{\text{n-1}}\left(y\right)}{{d\left(x\right)}^{n-1}}+\cdots +p_{n-1}\left(x\right)\frac{d\left(y\right)}{d\left(x\right)}+p_n\left(x\right)y\left(x\right)=g\left(x\right)$

If $g\left(x\right)=0$ then it is homogeneous.

If $g\left(x\right)\ne 0$ then it is non-homogeneous.

$y_{\mathrm{gen}}\left(x\right)=y_{\mathrm{hom}}\left(x\right)+y_{\mathrm{par}}\left(x\right)$

If the Wronskian is zero then the solutions aren't linearly independent:

$W(y_1,\cdots ,y_n)=\left|\begin{array}{cccc}{y}_1& y_2& \cdots & y_n\\ y_1^{\prime }& y_2^{\prime }& \cdots & y_n^{\prime }\\ & & & \\ {y_1}^{n-1}& {y_2}^{n-1}& \cdots & {y_n}^{n-1}\end{array}\right|$

If $k=\lambda +i\mu$ then:

$y_{\mathrm{hom}}=e^{\lambda x}(c_1\cos \left(\mu \right)x+c_2\sin \left(\mu \right)x)$

The Chain Rule:

$\frac{\partial \left(u\right)}{\partial \left(t\right)}=\frac{\partial \left(u\right)}{\partial \left(x\right)}\frac{\partial \left(x\right)}{\partial \left(t\right)}+\frac{\partial \left(u\right)}{\partial \left(y\right)}\frac{\partial \left(y\right)}{\partial \left(t\right)}$

Exact ODEs

$M(x,y)+N(x,y)\frac{d\left(y\right)}{d\left(x\right)}=0$

$\iff$

${\psi }_x=\frac{\partial \left(M\right)}{\partial \left(y\right)}=\frac{\partial \left(N\right)}{\partial \left(x\right)}={\psi }_x$

ODE Methods & Laplace Transforms -> click

D'Alembert: $y\left(x\right)=$old solution$\times u\left(x\right)$.

Variation of parameters: $y_{\mathrm{par}}={\sum }_{j=1}^n(u_j\left(x\right)y_j\left(x\right))$

Diagonalization: $x=Ty$:

$T{y}^{\prime }=ATy+g\left(t\right)\Rightarrow \frac{d\left(y\right)}{d\left(t\right)}=Dy+T^{-1}g=Dy+h$

Systems, Stability, Fourier, PDEs -> click

First solution:

$x^{\left(1\right)}=e^{\lambda t}{\xi }_{\lambda }$

Second solution:

$x^{\left(2\right)}=t{ℇ}^{\lambda t}{\xi }_{}+{ℇ}^{\lambda t}\eta$

where ${\xi }_{}=(A-\lambda 𝕀)\eta$. Third solution:

$x^{\left(3\right)}=\frac{t^2}{2}{e}^{\lambda t}{\xi }_{\lambda }+t{ℇ}^{\lambda t}\eta +e^{\lambda t}\zeta$

where $\eta =(A-\lambda 𝕀)\zeta$.

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