DE (x-2y) dx + y dy = 0
The given equation is a first-order linear differential equation, which can be solved by using an integrating factor. Let's learn how to solve it step by step:
Step 1: Rewrite the differential equation in a standard form
To start, let's rearrange the equation to have the differential terms on one side and constants on the other. The given equation is:
DE: (x - 2y) dx + y dy = 0
Rearranging, we have:
(x - 2y) dx + y dy = 0
x dx - 2y dx + y dy = 0
x dx + y dy - 2y dx = 0
Step 2: Identify the coefficients
From the rearranged equation, we can identify the coefficients of dx and dy:
Coefficient of dx: x + (-2y) = x - 2y
Coefficient of dy: y
Step 3: Determine the integrating factor
To find the integrating factor, we will compute the exponent of the integrating factor using the coefficients from the previous step. The integrating factor is given by:
Integrating factor (IF) = e^(∫P dx)
where P is the coefficient of dx.
In this case, the coefficient of dx is (x - 2y), so we have:
IF = e^(∫(x - 2y) dx)
To find the integral, we integrate with respect to x while treating y as a constant:
IF = e^(∫(x - 2y) dx)
= e^(x^2/2 - 2xy + C)
where C is the constant of integration.
Step 4: Multiply the differential equation by the integrating factor
Multiply the entire differential equation by the integrating factor (IF) determined in the previous step:
IF * (x dx + y dy - 2y dx) = 0
IF * x dx + IF * y dy - IF * 2y dx = 0
Step 5: Simplify and integrate
After multiplying the differential equation by the integrating factor, simplify and integrate both sides:
IF * x dx + IF * y dy - IF * 2y dx = 0
x(IF) dx + y(IF) dy - 2y (IF) dx = 0
Now, we can integrate both sides:
∫[x(IF) dx] + ∫[y(IF) dy] - ∫[2y (IF) dx] = ∫[0]
Step 6: Solve the integrals
Computing the integrals, we have:
∫[x(IF) dx] + ∫[y(IF) dy] - ∫[2y (IF) dx] = ∫[0]
Expanding the terms and solving each integral:
∫[x(IF) dx] = ∫[e^(x^2/2 - 2xy + C) dx]
= ... (integral of e^(x^2/2 - 2xy + C) with respect to x)
∫[y(IF) dy] = ∫[y * e^(x^2/2 - 2xy + C) dy]
= ... (integral of y * e^(x^2/2 - 2xy + C) with respect to y)
∫[2y (IF) dx] = ∫[2y * e^(x^2/2 - 2xy + C) dx]
= ... (integral of 2y * e^(x^2/2 - 2xy + C) with respect to x)
Note: Solving these integrals involves integration techniques such as u-substitution or integration by parts, depending on the specifics of each integral. The steps for solving these integrals can be quite involved and are beyond the scope of this explanation.
Step 7: Apply the initial condition (if given)
If there is an initial condition given, substitute the appropriate values into the general solution obtained from the integrals. This will determine the specific solution to the differential equation.
That's the general process to solve the first-order linear differential equation (x - 2y) dx + y dy = 0. Keep in mind that the steps can vary slightly depending on the specific equation and coefficients involved.
If you have a particular initial condition or need help solving the integrals, please provide those details and I'll be happy to assist you further!