# MATH128 Lecture Notes - Lecture 1: Switch, Product Rule

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School
Department
Course Math 128 Final Review
Chapter 9
9.1 Ordinary Differential Equations
Linear
Initial Value Problem (provide y, y’ or y^2) can ﬁnd particular solution
Separable ODE:
Particular solution: separate dy and y to the left and dx and x to the right,
then integrate. Make sure to check initial position when x = 0 don't divided
by zero
General solution: cant solve for c, don't need to show y=…
Make sure to check for y=0, if 0=0 then another solution
Use substitution to make a separable (Feb 3)
First Order Linear (Not Separable)
General Form: A(x)y’ + B(x)y = C(x) then manipulate to ->
Y’ + P(X)Y = Q(X), U(X) = e^integral of P(X)dx
Then multiply the whole ODE by U(X)
LHS becomes product rule, so integrate both sides
Solve for y and C
Mathematical Modelling
Mixing Problem
Different rate in and rate out rate. (dx/dt = rate of salt in - rate of salt
out) = (kg/L * ﬂows in rate)-[(amount of salt x / volume-drains out rate
+ﬂows in rate) * (drains out rate)]
Same rate in and rate out (separable), x/volume only since drain out
and ﬂow in rates are the same
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