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Math 3195
Linear Algebra and Differential Equations
Cheat Sheet
Name
Form of Equation
Separable
dy
dx
= f (x)
dy
dx
=
g(x)
f (y)
Solution Method
y(x) =
R
R
f (x)dx + C
f (y)dy =
R
Set ρ(x) = e
First-Order Linear
dy
dx
+ P (x)y = Q(x)
R
g(x)dx + C
P (x)dx
dy
So ρ(x) dx
+ ρ(x)P (x)y = ρ(x)Q(x).
Then ρ(x)y(x) =
R
ρ(x)Q(x)dx + C.
Substitutions
Let v = ax + by + c, solve for y,
dy
dx
= F (ax + by + c)
find
dy
dx
in terms of
dv
dx
and rewrite the equation in terms of
Let v =
dy
dx
Homogeneous
= F ( xy )
y
x
so y = vx and
dy
dx
dv
dx .
dv
= v + x dx
.
Rewrite equation in terms of
dv
dx .
If n = 0 or n = 1, this is linear, otherwise
Bernoulli
dy
dx
+ P (x)y = Q(x)y n
Let v = y 1−n , Apply the substitution to get a linear form.