Chapter 8: Ordinary Differential Equations I. General A linear ODE is of the form: An n th order ODE has a solution containing n arbitrary constants ex: Ch. 8- Ordinary Differential Equations > General

Three really common ODE’s: 1) 2) 3) Ch. 8- Ordinary Differential Equations > General

How do we solve for the constants? → In general, any constant works. → But many problems have additional constants (boundary conditions) and in this case, the particular solution involves specific values of the constants that satisfy the boundary condition. ex: for t<0, the switch is open and the capacitor is uncharged. at t=0, shut switch Ch. 8- Ordinary Differential Equations > General

II. Separable Ordinary Differential Equations A separable ODE is one in which you can separate all y-terms on the left hand side of the equation and all the x-terms on the right hand side of the equation. ex: xy’=y We can solve separable ordinary differential equations by separating the variables and then just integrating both sides Ch. 8- Ordinary Differential Equations > Separable Ordinary Differential Equations

ex: xy’=y subject to the boundary condition y=3 when x=2 Ch. 8- Ordinary Differential Equations > Separable Ordinary Differential Equations

ex: Rate at which bacteria grow in culture is proportional to the present. Say there are no bacteria at t=0. subject to boundary condition N(t=0)=N o Ch. 8- Ordinary Differential Equations > Separable Ordinary Differential Equations

ex: Schroedinger’s Equation: solve for the wave function if V(x,t) is only a function of x, e.g. V(x), then schroedinger’s equation is separable. Ch. 8- Ordinary Differential Equations > Separable Ordinary Differential Equations

III. Linear First-Order Ordinary Differential Equations Definition: a linear first-order ordinary differential equation can be written in the form: y’+Py=Q where P and Q are functions of x the solution to this is: Check: Ch. 8- Ordinary Differential Equations > Linear First-Order ODEs

ex: Ch. 8- Ordinary Differential Equations > Linear First-Order ODEs

IV. Second Order Linear Homogeneous Equation A second order linear homogeneous equation has the form: where a 2, a 1, a 0 are constants To solve such an equation: let Ch. 8- Ordinary Differential Equations > Second-Order Linear Homogeneous Equation

ex: y’’+y’-2y=0 Ch. 8- Ordinary Differential Equations > Second-Order Linear Homogeneous Equation

ex: Harmonic Oscillator