18.03 Lecture Notes - Lecture 2: Linear Form, Product Rule
Recitation Handout
Review: a homogeneous solution is equal to zero, and so it is easier to solve
!Homogeneous: Inhomogeneous:
a) !Standard Linear form is y' + xy =0
!Linear means that y is not raised to a power greater than 1
!The equation y' - (tanx)y =1 is linear but not separable because it is homogenous
b)!To solve for a basic solution, simply set the equation equal to zero and solve using separation of variable
c)!Plug in uy(h) for y:
!Use the product rule to expand:
!Plug in for y(h) and y(h)':
!Simplify:
d)!We multiply the equation by cosx:
!Use separation of variables:
a)!
!Polar coordinates are (r, θ):
!We can also write this as
Document Summary
Review: a homogeneous solution is equal to zero, and so it is easier to solve. Standard linear form is y" + xy =0. Linear means that y is not raised to a power greater than 1. The equation y" - (tanx)y =1 is linear but not separable because it is homogenous. To solve for a basic solution, simply set the equation equal to zero and solve using separation of variable. Use separation of variables: we multiply the equation by cosx:
