Right now my class is working on exponential growth and decay. I know that i have to find the integral of both sides and solve for Y but I don't know how to do so exactly.
Here's the problem:
if dy/dx=3Y(10-Y)
integrate to solve for Y
y is the integral of dy/dx. I assume that all your y's should be lower case.
You have a differential equation that can be solved by separation of variables.
Integral of dy/[3y(10-y)] = x
Do the integral and you will have an equation for x in terms of y. There will be an arbitrary constant in the answer, unless you integrate between specific limits and know an initial condition.
To solve the given differential equation and find the function Y(x), you can use separation of variables and integrate both sides of the equation. Here's how you can proceed:
Step 1: Rewrite the given differential equation in the form dy/dx = f(x)g(y):
dy/dx = 3y(10 - y)
Step 2: Separate the variables by multiplying both sides by dx and dividing by g(y):
1/(y(10-y)) dy = 3 dx
Step 3: Integrate both sides with respect to their respective variables.
Integrate the left side with respect to y:
∫ 1/(y(10-y)) dy
This integral can be solved by using partial fractions decomposition or by using a substitution. Let's use the method of partial fractions.
Step 3a: Factor the denominator:
y(10-y) = y(10-y) = y(10 - y)
Step 3b: Perform partial fraction decomposition:
1/(y(10-y)) = A/y + B/(10-y)
To find the values of A and B, you can multiply each fraction by the denominator of the other term and then equate the numerators:
1 = A(10-y) + By
Simplify the equation:
1 = 10A - Ay + By
To solve for A and B, let's assign some values to y:
If y = 0, the equation becomes:
1 = 10A
So, A = 1/10
If y = 10, the equation becomes:
1 = 10B
So, B = 1/10
Now we can rewrite the original integral as:
∫ 1/(y(10-y)) dy = ∫ (1/10) * (1/y) dy + ∫ (1/10) * (1/(10-y)) dy
Step 3c: Perform the integrals:
∫ (1/10) * (1/y) dy = (1/10) * ln|y| + C1
∫ (1/10) * (1/(10-y)) dy = (1/10) * ln|10-y| + C2
Where C1 and C2 are constants of integration.
Step 4: Integrate the right side of the original equation:
∫ 3 dx = 3x + C3
Where C3 is another constant of integration.
Step 5: Combine the integrals and constants:
(1/10) * ln|y| + (1/10) * ln|10-y| = 3x + C
Here, C = C3 - (C1 + C2)
Step 6: Simplify the logarithmic terms:
ln|y|/10 + ln|10-y|/10 = 3x + C
Step 7: Combine logarithmic terms into a single logarithm:
ln|y(10-y)|/10 = 3x + C
Step 8: Remove the natural logarithm by taking the exponential of both sides:
e^(ln|y(10-y)|/10) = e^(3x + C)
y(10-y) = e^(10(3x + C))
Step 9: Simplify the exponential expression:
Since e^C is a constant, let's denote it as K:
y(10-y) = Ke^(30x)
Step 10: Rearrange the equation and solve for y:
10y - y^2 = Ke^(30x)
This is a quadratic equation, so let's solve it by setting it equal to 0 and applying the quadratic formula:
y^2 - 10y + Ke^(30x) = 0
The quadratic formula is:
y = (-b ± √(b^2 - 4ac)) / (2a)
For our quadratic equation, a = 1, b = -10, and c = Ke^(30x). Substitute these values into the formula and simplify as needed to find the solutions for y.
Remember that the ± sign in the quadratic formula indicates that there are two possible solutions for y: one positive and one negative.
Once you've found the solutions for y, you can write the general solution by combining the equations from the quadratic formula and determining the values of the integration constants C1, C2, and C3.