Math 2142 Homework 4, Linear ODE Solutions
Math 2142 Homework 4, Linear ODE Solutions
Problem 1. Consider the first initial value problem.
dy = (1 - 2x)y2 with y(0) = -1/6 dx Rewriting this separable equation, we get
1 y2 dy = (1 - 2x) dx
-1 = x - x2 + c y
Plugging in y = -1/6 and x = 0 from the initial condition gives 6 = c. Solving for y gives
-1 = x - x2 + 6 y 1 = -x + x2 - 6 y
1 y = x2 - x - 6
Now consider the second initial value problem.
dy = xy3(1 + x2)-1/2 dx
with
y(0) = 1
Rewriting this separable equation, we get
1 dy =
y3 -1 = 2y2
x
dx
1 + x2
x
dx
1 + x2
To do the remaining integral, use the substitution u = 1 + x2, so du = 2x dx. This gives
x
1 dx =
1
du = u + c = 1 + x2 + c
1 + x2
2u
Therefore, altogether we have
-1
=
1 + x2
+c
2y2
Plugging in y = 1 and x = 0 from the initial condition gives -1/2 = 1 + c, so c = -3/2. To solve for y
-1
3
2y2 =
1 + x2 - 2
1
y2
=
-2
1 + x2 + 3
y = (3 - 2 1 + x2)-1/2
Problem 2. For the first differential equation
dy
x2
=
dx y(1 + x3)
we separate the variables to get
x2
y dy =
dx
1 + x3
The y integral is just y2/2. To do the x integral, use the substitution u = 1+x3, so du = 3x2 dx.
x2
1
dx =
1
du
=
1
ln |u|
+
c
=
1
ln |1
+
x3|
+
c
=
ln( 3 1
+
x2)
+
c
1 + x3
3u
3
3
We can remove the absolute value signs because x > 0. Altogether, we have
y2/2 = ln( 3 1 + x2) + c
y = ? 2 ln( 3 1 + x2) + c
where the c in the second line is really twice the constant from the first line. For the second differential equation
dy
2x
=
dx yey - x2yey
we separate the variables to get
yey dy =
2x dx
1 - x2
To calculate the y integral, we use integration by parts. Let u = y and dv = ey dy. Then du = dy and v = ey, giving us
yey dy = yey - ey dy = yey - ey
(and I'll save the integration constant for the end). To do the x integral, use the substitution u = 1 - x2, so du = -2x dx.
2x dx = -
1 du = - ln |1 - x2| + c = - ln(1 - x2) + c
1 - x2
u
Note that we can remove the absolute value signs because -1 < x < 1. Altogether, we get
yey - ey = - ln(1 - x2) + c
which we will not attempt to solve for y!
Problem 3(a). Solve the following algebraic equation explicitly for y. y2 - 2y + 4 = x4 + x2 + 6
Solution. As indicated on the homework, we completing the square on the left side and solve for y.
(y - 1)2 - 1 + 4 = x4 + x2 + 6 (y - 1)2 + 3 = x4 + x2 + 6 (y - 1)2 = x4+ x2 + 3 y = ? x4 + x2 + 3 + 1
3(b). Solve the following separable initial value problem and give the solution explicitly as a
function y(x).
dy 3x2 + ex
=
with y(0) = 1
dx 2y - 4
Solution. Separating the variables, integrating and completing the square gives
2y - 4 dy = 3x2 + ex dx
y2 - 4y = x3 + ex + c (y - 2)2 - 4 = x3 + ex + c
(y - 2)2 = x3 +ex + c y = 2 ? x3 + ex + c
where c = c + 4 is an altered constant of integration. To solve for the constant c, we plug in
y = 1 and x = 0 to get
1=2? 0+1+c
It follows that c = 0 and we need to use the negative square root. So, the specific solution is
y = 2 - x3 + ex
Problem 4. A tank contains 1000 liters of water with 15 kg of dissolved salt. Pure water enters the tank at a rate of 10 liters per minutes and the mixture is drained from the bottom at the same rate. Assume that the solution is kept thoroughly mixed, how many kilograms of salt are in the tank after t minutes?
Solution. Let y(t) be the amount of salt (in kilograms) in the tank after t minutes. We are given y(0) = 15. To express dy/dt, we need to determine the rates at which salt is entering
and leaving the tank. Since the water entering the tank contains no salt, the rate at which salt enters is 0. The rate at which salt is leaving the tank is
y kg
Ly
? 10 =
1000 L min 100
measured in kilograms per minute. Therefore, our initial value problem for this situation is
dy
y
=-
with y(0) = 15
dt 100
We know that the solution to a differential equation of the form dy/dt = ky is y = Aekt where A = y(0). Therefore, the solution here is
y(t) = 15e-t/100
Problem 5. Newton's Law of Cooling states that the rate of cooling (or heating) of an object is proportional to the temperature difference between the object and its surroundings. In other words, if T (t) is the temperature of the object at time t and Ts is the constant temperature of its surroundings, then Newton's Law of Cooling says that T (t) satisfies
dT dt = k(T (t) - Ts)
for some constant k.
5(a). A can of soda at 72 degrees is placed in a refrigerator where the temperature is 44 degrees. Use Newton's Law of Cooling to set-up and solve a differential equation for T (t), the temperature of soda after t minutes. (The constant k will appear in your answer but you can solve for the constant of integration using the initial temperature of the can.)
Solution. Let T (t) be the temperature of the can t minutes after it is placed in the refrigerator. Since the temperature of the refrigerator is 44 degrees, we have Ts = 44. Since the initial temperature of the can is 72 degrees, we have T (0) = 72. Therefore, our initial value problem is
dT = k(T - 44) with T (0) = 72
dt Separating the variables gives
1 dT = k dt
T - 44 ln(T - 44) = kt + c
T - 44 = Aekt T (t) = Aekt + 44
where A = ec is the altered constant of integration. Note that we do not need absolute value signs for ln |T - 44| because T (0) = 72, so we know T (t) - 44 is positive. To find the constant
A of integration, we plug in T = 72 and t = 0 to get 72 = A + 44, or in other words, A = 28. Altogether, our solution looks like
T (t) = 28ekt + 44
5(b). After 20 minutes, the temperature of the can is 61 degrees. Use this information to find the constant k.
Solution. We are given T (20) = 61, so we can find k by plugging in t = 20 and T = 61. 61 = 28e20k + 44
which means k = -0.025 (approximately). 5(c). How long does it take for the soda to cool to 50 degrees?
Solution. We need to solve
50 = 28e-0.025t + 44
which gives t = 62 (approximately).
Problem 6. One model for the spread of a rumor is that the rate of spread is proportional to the product of the fraction of people who have heard the rumor and the fraction of the people who have not heard the rumor. If y(t) denotes the fraction of people who have heard the rumor after t days, then the modeling differential equation is
dy = ky(1 - y)
dt
where k is the proportionality constant.
6(a). Find the general solution to this separable differential equation. (You can assume that 0 < y < 1, and hence 0 < 1 - y < 1 as well. The constant k will appear in your solution in addition to the constant of integration.)
Solution. Separating the variables gives
1 dy = k dt
y(1 - y)
The t integral is just kt + c, but to do the y integral, we need to use partial fractions.
1
A B A(1 - y) + By
=+
=
y(1 - y) y 1 - y
y(1 - y)
and therefore, 1 = A(1 - y) + By. Since this equality has to hold for all y, we can plug in y values to help us determine the constants A and B. Plugging in y = 1 gives us 1 = B and plugging in y = 0 gives us 1 = A. So, our partial fraction decomposition is
1
11
=+
y(1 - y) y 1 - y
................
................
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