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Explore linear and separable ODEs, solve for constants, and tackle second-order equations. Learn to apply principles to real-world scenarios. Dive into the concepts of Schroedinger's Equation and the Harmonic Oscillator.
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Ch. 8- Ordinary Differential Equations > General • Chapter 8: Ordinary Differential Equations • I. General • A linear ODE is of the form: • An nth order ODE has a solution containing n arbitrary constants • ex: • 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 > Separable Ordinary Differential Equations 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)=No
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 > Linear First-Order ODEs 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 > Second-Order Linear Homogeneous Equation IV. Second Order Linear Homogeneous Equation A second order linear homogeneous equation has the form: where a2, a1, a0 are constants To solve such an equation: let
Ch. 8- Ordinary Differential Equations > Second-Order Linear Homogeneous Equation ex: y’’+y’-2y=0
