Linear ODE

Description

Eine lineare Differentialgleichung ist eine Ordinary differential equation, die Linear von den betrachteten Variablen und ihren Ableitungen abhängt.

Zur Behandlung von mehrdimensionalen ODEs, siehe System of linear ODEs

Definition

Eine Ordinary differential equation wird linear genannt, wenn jede Gleichung in die Form $x^{(i)}=a_0(t)x+...+a_{i-1}(t)x^{(i-1)}+b_0(t)y+...+v(t)$ umgewandelt werden kann.

Die Existenz des $v(t)$ entscheidet darüber, ob das gegebene System homogen oder inhomogen ist

Properties

Solution of first-order homogenous linear ODEs

If the linear ODE is homogenous, meaning it is of the form $x^{\prime }(t)=a(t)b(x),x(t_0)=x_0$ the solution can be calculated using Separation of variables and is $\lambda (t)=x_0{e}^{{\int }_{t_0}^{t}a(s)ds}$

Solution of first-order nonhomogenous linear ODEs

If the linear ODE is inhomogenous, meaning it is of the form $x^{\prime }(t)=a(t)b(x)+b(t),x(t_0)=x_0$ the solution can be calculated using Variation of parameters and is $\lambda =x_0{e}^{A(t)}+e^{A(t)}\underset{t_0}{\overset{t}{\int }}b(s)e^{-A(s)}ds$ Or as I like to put it (in terms of the homogenous solution ${\lambda }_h$): $\lambda (t)={\lambda }_h(t)+{\lambda }_h(t)\underset{t_0}{\overset{t}{\int }}b(s){\lambda }_h(s{)}^{-1}ds\cdot$

Reduction to dimensionality

Linear ODE of higher order can be broken down into a System of linear ODEs of higher dimension. The equation $y^{(n)}+a_{n-1}{y}^{(n-1)}+...+a_1{y}^{\prime }+a_{0}y=0$ can be transformed by using the new variables $u_0=y,u_1=y^{\prime },...,u_n=y^{(n)}$ giving us

u_{1}' &= u_{2}\\ \vdots &= \vdots\\ u_{n-2}' &= u_{n-1}\\ u_{n-1}' &= -a_{n-1}u_{n-1} - ... -a_{0}u_{0} \end{align}$$ Which is equal to $$\begin{pmatrix} u_{0}' \\ \vdots \\ u_{n-1}'\end{pmatrix} = \begin{pmatrix} 0 & 1 & ... & 0 \\ \vdots & \ddots & \ddots & \vdots \\ -a_0 & ... & -a_{(n-2)} & -a_{(n-1)} \end{pmatrix}\begin{pmatrix} u_{0} \\ \vdots \\ u_{n-1}\end{pmatrix}$$

From this follows that systems of higher order also have a superposition principle. Their solutions form a vector space. The points of the vectors space determine the initial position, the initial first derivative and so on. From this we get the following Fundamental system (ODE)

Fundamental system

Let there be a linear ODE of order $n$ like above. If $\rho$ is a $k$-times root of the characteristic polynomial of the matrix written above then the following are linearly independent solutions $e^{\rho t},t{e}^{\rho t},...,t^{k-1}{e}^{\rho t}$ and together with all other roots they form a Fundamental system (ODE).

Note: If the linear ODE is real-valued and $\rho$ is imaginary, e.g. $\rho =1+i$, then $1-i$ is a root too. $\frac{1}{2}(e^{(1+i)t}+e^{(1-i)t})=e^{t\cos (t)}\text{ and }\frac{1}{2i}(e^{(1+i)t}-e^{(1-i)t})=e^{t\sin (t)}$ then is a pair of real solutions.

Examples

Even though higher order linear ODEs can be transformed into higher-dimensional ODEs the former is much easier, since we can write them in one dimension. I guess, they just have way more structure. This allows us to more easily find a correct function by guessing.

Linear ODEs solvable by an ansatz

The last coefficient $v(t)$ imposes hard restriction on what solutions are allowed. This allows us to use an Ansatz-strategy (similar to Variation of parameters): Guess a parametrised solution and determine the parameters.

Let the ODE be of the form $x^{(i)}+a_0(t)x+...+a_{i-1}(t)x^{(i-1)}+b_0(t)y+...=v(t)$

Right side of the ODE	Parameter	Ansatz
$e^{\alpha t}$	$\alpha$	$C{t}^k{e}^{\alpha t}$
$p_l(t)e\alpha t$	$\alpha$	$t^k{r}_l(t)e^{\alpha t}$
$A\sin (\beta t)+B\cos (\beta t)$	$\beta$	$t^k(C\sin (\beta t)+D\cos (\beta t))$
$(p_l(t)A\sin (\beta t)+q_l(t)B\cos (\beta t))e^{\alpha t}$	$\alpha ,\beta$	$t^k(r_l(t)A\sin (\beta t)+s_l(t)B\cos (\beta t))e^{\alpha t}$
$p_l,q_l,r_l$ und $s_l$ bezeichnen Polynome von Grad $l$.