solve dy/dx = y^2 - 4 using Separation of Variables?
To solve the differential equation dy/dx = y^2 - 4 using separation of variables, follow these steps:
Step 1: Separate the variables
Write the differential equation in the form of dy/dx = f(x)g(y). In this case, we have dy/dx = (y^2 - 4).
dy/(y^2 - 4) = dx
Step 2: Integrate both sides
Integrate both sides of the equation with respect to their respective variables.
∫ dy/(y^2 - 4) = ∫ dx
Step 3: Evaluate the integrals
The integral ∫ dy/(y^2 - 4) can be evaluated using partial fractions. The denominator can be factored as (y - 2)(y + 2):
1/[(y - 2)(y + 2)] = A/(y - 2) + B/(y + 2)
Multiplying through by (y - 2)(y + 2), we get:
1 = A(y + 2) + B(y - 2)
Solving for A and B by equating the coefficients, we find A = 1/4 and B = -1/4.
Now, we can integrate:
∫ (1/[(y - 2)(y + 2)]) dy = ∫ dx
(1/4) ∫ (1/(y - 2)) dy - (1/4) ∫ (1/(y + 2)) dy = ∫ dx
(1/4) ln|y - 2| - (1/4) ln|y + 2| = x + C
Step 4: Solve for y
Combine the logarithmic terms on the left side:
ln|y - 2| - ln|y + 2| = 4x + 4C
Using the properties of logarithms, we can simplify:
ln|y - 2|/(y + 2) = 4x + 4C
Exponentiate both sides:
|y - 2|/(y + 2) = e^(4x + 4C)
Step 5: Remove the absolute value
Since the absolute value notation is not convenient, we can consider both cases:
Case 1: y - 2 > 0 (y > 2)
In this case, the absolute value becomes:
(y - 2)/(y + 2) = e^(4x + 4C)
Simplify the equation:
y - 2 = (y + 2)e^(4x + 4C)
Distribute the exponential term:
y - 2 = (y + 2) * e^(4x) * e^(4C)
Since e^(4C) is a constant, we can combine it with y + 2:
y - 2 = Ae^(4x)
Case 2: y - 2 < 0 (y < 2)
In this case, the absolute value becomes:
-(y - 2)/(y + 2) = e^(4x + 4C)
Simplify the equation:
- (y - 2) = (y + 2)e^(4x + 4C)
Distribute the exponential term:
- y + 2 = (y + 2) * e^(4x) * e^(4C)
Simplifying further:
- y + 2 = Ae^(4x)
Step 6: Combine both cases
Combining both cases, we have:
y - 2 = Ae^(4x)
or
- y + 2 = Ae^(4x)
These are two possible solutions to the differential equation.