Differential Equations

Question 1

What is the order of a differential equation

Answer 1

The highest order of derivative. Dictates number of arbitrary constants

Question 2

What is the degree of a differential equation

Answer 2

The highest power to which a derivative is raised

Question 3

What is the general expression for a separable equation

Answer 3

dy/dx=f(y)g(x)

Question 4

What are the equations and rules for use of the integrating factor

Answer 4

Use if first order ODE takes inhomogeneous form:

dy/dx+ P(x)y = Q(x)

y=e^-I ∫Q(x)e^I dx +Ce^-I

where I = ∫P(x)dx (integrating factor)

Question 5

What is linearity

Answer 5

If we have two solutions y1 and y2, then a linear combination of both is also a solution (with some constant)

eg. y(x)=Ay1(x)+By2(x)

Question 6

what are two equivalent solution forms for second order ODEs with complex conjugate solutions

Answer 6

y(x)=e^xmu(Ccos(kx)+Dsin(kx))
[C=A+B and D=i(A-B)]

equivalent: y(x)=A0e^xmu*cos(kx+phi)

Question 7

how to find a particular solution for an inhomogeneous second order ODE

Answer 7

*if = constant: divide co-eff of the y term to get PS
*if = f(x), PS is in same form as this function. Sub in a function form and compare co-effs (remember to include all possible co-efficient forms)