Summary Separable Differential Equations | Calculus II Notes

2.4 Separable Differential Equations
an equation involving a function and some of its derivative is called a differential equation
the order of a differ, is the order ofthe heighest derivative equation
appearing in the

eg.
f" f
(x): (x) is a fourth order




solving a
differ find all functions thatsatisfy the equation.



example:y':f(x)


y Sf(x)dX
=




=x 2

y
y
(x'dx
=
=
c
+




there are many solutions depending on the C value

for a particular solution pick outa Cvalue of specify a
pointon y= f(x), or specify a value on f




example:

findy, x2
y (1)
3
y' and
= =




firstsolve the integral
y (x'dx
c
2
= +
=




the solve for cat y(1) 3
=




bz c 3
+
=




C 3 =
-

1z 8/3
=




y x3x 83
+
=




specifying a value off is called an initial condition




first order separable diff.egs.


form: f(x). g(y)
1
=




steps to solve:
1)"separate"the variables


giys dy f(x)dx
=




-)

integratebedee