2. Separable Differential Equations
A separable differential equation can be expressed as the product of a
function of x and a function of y.
dy
g x h y h y 0
dx
Example:
Multiply both sides by dx and divide both sides by y2
dy
2 xy 2 to separate the variables. (Assume y2 is never zero.)
dx
dy
2 x dx
y2
y 2 dy 2 x dx
3. Separable Differential Equations
A separable differential equation can be expressed as the product of a
function of x and a function of y.
dy
g x h y h y 0
dx
Example:
y 2 dy 2 x dx
dy
2 xy 2 -1
dx -y = x +C 2
dy 1
2 x dx x2 C
y2 y
1
y 2 dy 2 x dx y
x2 C
4. Example:
Solve the differential with initial condition
dy 2 x2
2x 1 y e Separable differential equation
dx
1 x2
2
dy 2 x e dx
1 y
1
dy 2 x e dx x2 u x2 du = 2x dx
2
1 y du
= dx
1 2x
2
dy eu du
1 y
tan -1 y = e u + C
-1
tan y = e + C x2
5. Example (cont.):
dy 2 x2
2x 1 y e
dx
1 x2
tan y e C We now have y as an implicit function of x.
1 x2
tan tan y tan e C We can find y as an explicit function of x by
taking the tangent of both sides.
x2
y tan e C
6.
7. A harder problem
Careful when
separating, we
can only separate
by multiplying
or dividing!!!
