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Unformatted text preview: MA 36600 LECTURE NOTES: FRIDAY, JANUARY 23 First Order Nonlinear Equations (cont’d)
Example. Say we wish to solve the initial value problem
3 t2 + 4 t + 2
2 (y − 1) y (0) = −1. First we ﬁnd the general solution to the diﬀerential equation, then we determine the constants of integration
from the initial condition.
This diﬀerential equation is separable because we have
= Y (y ) T (t)
dt where Y (y ) = 1
2 (y − 1) T (t) = 3 t2 + 4 t + 2. We can bring all of the y terms to one side, and all of the t terms to the other:
2 (y − 1)
= 3t + 4t + 2 .
The left-hand side is
2 (y − 1)
y − 2y
y − 2 y = 3 t2 + 4 t + 2 .
Upon integrating both sides, we see that
y 2 − 2 y = t3 + 2 t2 + 2 t + C for some constant C . These are the level curves. For the initial condition, set t = 0:
C = y (0)2 − 2 y (0) = (−1)2 − 2 · (−1) = 3. Hence the solution to the initial value problem is the implicit solution
y 2 − 2 y = t3 + 2 t2 + 2 t + 3 . The graph of this solution is called an elliptic curve. Modeling with First Order Equations
We give several examples of ﬁrst order equations by discussing several types of mathematical models.
Mixing Problems. Consider a tank which contains 100 gallons of water. A salt-water solution containing
1/4 lbs of salt per gallon is entering the tank at a constant rate, and a faucet is opened at the bottom of the
tank so that the total amount of water in the tank remains constant. What is the amount of salt (in lbs)
after a long time passes?
We will solve this problem two diﬀerent ways, but ﬁrst we ﬁnd a diﬀerential equation which models the
situation. Let Q(t) denote the quantity of salt in lbs at time t. We wish to compute the “limiting quantity”
QL = lim Q(t).
t→∞ We do not have all of the variables needed to write down the diﬀerential equation, so we introduce those
variables. Say that initially there is Q(0) = Q0 lbs of salt dissolved in the water, and that the salt-water
solution enters the tank at a rate of r gallons per minute. We form the following table: Flowing In
Flowing Out Concentration of Salt Rate of Flow
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