Differential Equations Summary
We give a summary of our results about differential equations in the form of a flow chart. We will
assume that first order differential equations are either linear or separable and that second order
differential equations are homogeneous and linear with constant coefficients.
Given:
Differential Equation
First or
Second
Order?
Linear?
Put in Standard Form:
d2y
dx2+pdy
dx +qy = 0
Solve Auxiliary Eq:
m2+pm +q= 0
Put in Standard Form:
dy
dx +p(x)y=q(x)
Put in the Form:
dy
dx =h(x)g(y)
Set Integrating Factor:
µ=eRp(x)dx
General Solution:
y(x) = 1
µ(x)Zµ(x)q(x)dx
Rewrite as:
1
g(y)dy =h(x)dx
Integrate both sides:
Z1
g(y)dy =Zh(x)dx
Let G, H be
antiderivatives
for 1/g, h, respec-
tively
General Solution
(solve for yif possible):
G(y) = H(x) + C
Type of
Roots?
General Solution:
y(x) = c1em1x+c2em2x
General Solution:
y(x) = c1emx +c2xemx
General Solution:
y(x) = eax[c1cos(bx)+c2sin(bx)]
1st order
2nd order
yes
no
distinct real
roots m1, m2
double root m
complex roots a±bi