Chapter 4 (a&b) Flashcards
Second-Order Linear ODE - General Form
e(x)d²y/dx² + f(x)dy/dx+ g(x)y
=h(x)
Homogeneous and Inhomogeneous Second-Order Linear ODEs
-a second order linear ODE is homogeneous if h(x)=0
-it is inhomogeneous if
h(x) ≠ 0
Constant Coefficient and Non-Constant Coefficient Second-Order Linear ODEs
- a second order linear ODE is constant coefficient if e(x), f(x) and g(x) are all constants, rather than functions of x
- otherwise the equation is non-constant coefficient
Homogeneous Second Order Linear ODEs
Initial Value Problem
-given values of y and y’ at a single value of x
Homogeneous Second Order Linear ODEs
Boundary Value Problem
-given the values of y at two different values of x
Second Order Linear ODEs
Theorem for Initial Value Problems
- let e(x), f(x), g(x) and h(x) be continuous on an interval I of the x-axis and let e(x) ≠ 0 for all x in I
- if the initial condition is specified at a point in I then the solution exists and is unique
Superposition Principle for Linear ODEs
Linear Superposition
- suppose that y1(x) and y2(x) are two solutions of the homogenous second order linear ODE
- then C1y1 + C2y2 is called a linear superposition of the two functions
Superposition Principe for Linear ODEs
Particular Integral and Complementary Functions
- suppose that y0(x) is a solution of the inhomogeneous second order linear ODE
- and y1(x) is a solution of the homogeneous form of the same equation
- then y0(x) + C1y1(x) is also a solution of the inhomogeneous form of the ODE
- y0 is the particular integral and C1y1, C2y2, or C1y1+c2y2 are complimentary functions
Reduction of Order
1) let y1(x) be a solution that we know to the homogeneous form of the equation, now let
y(x) = u(x) y1(x)
2) calculate the first and second derivatives of y
3) substitute y, y’ and y’’ into the inhomogeneous form of the equation and collect terms containing u, u’ and u’’
4) the coefficient of the u term should be the second order ODE solved by y1 which we know is homogeneous so equals 0
5) only terms in u’ and u’’ remain, use a substitution
v = u’ this turns the second order ODE into a first order ODE
6) solve for v by the integrating factor method
7) integrate v to find u remembering to add the constant of integration
8) substitute u into y=uy1 to find y
9) if given use any initial or boundary values to find the values of the constants
First Order Constant Coefficient Linear ODEs
General Form
dy/dx + ay = h(x)
First Order Constant Coefficient Linear ODEs
Complimentary Function
ycf = C e^(-ax)
General Solution, Complimentary Function, Particular Integral
-the general solution is the sum of the complimentary function and particular integral
y(x) = ycf + ypi
Complimentary Function
-the solution to the homogeneous form of the equation
Particular Integral
h(x) = constant
ypi = constant
Particular Integral
h(x) = polynomial
ypi = polynomial of the same degree as h(x)
-but if the h(x) is a polynomial times the complimentary function then ypi is a polynomial one degree higher than h(x)