PX153 - D1 - definitions

• a linear ODE can be written in the form:

- dependent variable and its derivatives are only present to first order (no $y^2$, etc)
- no non-linear functions of the dependent variable (no sin(y), etc)
- no products of the dependent variable and its derivatives (no $y\frac{dy}{dx}$, etc)

• a linear ODE is homogeneous if $f(x)=0$

  • then, if $y$ is a solution, so is $Ay$, or if $y_1$ and $y_2$ are both solutions, so is $A{y}_1+B{y}_2$
  • if $f(x)\ne 0$ and the ODE is linear, then it is inhomogeneous
  • eg:
    • $\sin x\frac{dy}{dx}=x-y$ - 1st order, linear, inhomogeneous
    • $\ddot{x}+\gamma \dot{x}+{\omega }_0^{2}x=0$ - 2nd order, linear, homogeneous
    • $\ddot{x}+\gamma \dot{x}+{\omega }_0^{2}x=b\cos \omega t$ - 2nd order, linear, inhomogeneous
    • $\ddot{x}+\gamma \dot{x}+x(x^2-1)=b\cos \omega t$ - forced Duffing's equation - 2nd order, non-linear

• the solution of an ODE is the relationship between the independent and dependent variable over a specified domain

• the general solution contains all solutions of the ODE

• for a linear homogeneous ODE, if $y_1(x)$ and $y_2(x)$ are solutions, so is $y(x)=A{y}_1(x)+B{y}_2(x)$

• eg: for $\ddot{x}+{\omega }_0^{2}x=0$

  • solutions: $x_1=A{e}^{i{\omega }_{0}t},x_2=B{e}^{-i{\omega }_{0}t}$
  • consider a constant times $x_1$, ie: $x=Cx$ :
  $$\ddot{x}+{\omega }_0^2=C{\ddot{x}}_1+C{\omega }_0^2{x}_1=C({\ddot{x}}_1+{\omega }_0^2{x}_1)=0$$
  • for $x=C{x}_1+D{x}_2$:
  $$\begin{array}{rl}\ddot{x}+{\omega }_0^{2}x& =\frac{d^2}{d{t}^2}(C{x}_1+D{x}_2)+{\omega }_0^2(C{x}_1+D{x}_2)\\ & =C({\ddot{x}}_1+{\omega }_0^2{x}_1)+D({\ddot{x}}_2+{\omega }_0^2{x}_2)\\ & =0\end{array}$$

  - so, $x$ is a solution
  - note: not true for inhomogeneous or non-linear

• for a second order ODE, expect two constants of integration

• the particular solution satisfies a given set of boundary conditions