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Differential Equations: Solving (x-1)dy - 2dx
In differential equations, solving equations of the form (x-1)dy - 2dx is a fundamental concept. This type of equation is known as a linear differential equation, and its solution is crucial in various fields such as physics, engineering, and mathematics.
The Equation: (x-1)dy - 2dx
The equation (x-1)dy - 2dx is a first-order linear differential equation. To solve this equation, we need to find the general solution, which is a function of x that satisfies the equation.
Method of Solution
To solve this equation, we can use the method of separation of variables. This method involves separating the variables x and y, and then integrating both sides of the equation with respect to x.
Step 1: Separate the Variables
First, we separate the variables x and y by dividing both sides of the equation by dx:
(x-1)dy = 2dx
Now, we have:
dy / (x-1) = 2dx / x
Step 2: Integrate Both Sides
Next, we integrate both sides of the equation with respect to x:
∫(1 / (x-1)) dy = ∫(2 / x) dx
Step 3: Evaluate the Integrals
Evaluating the integrals, we get:
ln|x-1| + C1 = 2ln|x| + C2
where C1 and C2 are constants of integration.
General Solution
Simplifying the equation, we get:
y = 2ln|x| - ln|x-1| + C
where C is the constant of integration.
Conclusion
In this article, we have solved the differential equation (x-1)dy - 2dx using the method of separation of variables. The general solution to this equation is y = 2ln|x| - ln|x-1| + C. This solution is essential in modeling various real-world phenomena, such as population growth, chemical reactions, and electrical circuits.
