Linear Equations
A linear first-order DE looks like
a1  x 
dy
dx
 a0  x  y  g  x 
Standard form is
dy
dx  P  x  y  Q  x 
To solve first-order linear DEs, we will be
P  x  dx

using the term e
, called the
integrating factor.
Solving first-order linear DE
 P x y  Q x
P  x  dx

2) Find the integrating factor e
1) Put into standard form
dy
dx
3) Multiply both sides of the DE by the
integrating factor. The result will be:
P  x  dx
d   P x dx 

e
y  Q xe

dx 
4) Integrate both sides of the equation.
Ex. Solve
dy
dx
 3y  0
Ex. Solve
dy
dx
 3y  6
Transient terms are terms that approach zero
as x goes to infinity.
Ex. Solve x
dy
dx
 4y  x e
6 x
What is the interval of definition?
Ex. Solve  x  9 
2
dy
dx
 xy  0
Ex. Solve
dy
dx
 y  x, y  0   4
Ex. Solve
dy
dx
 2 xy  2, y  0   1
 y  f  x  , y  0   0, where
1 0  x  1
f  x  
x 1
0
Ex. Solve
dy
dx