Solving the Differential Equation Using Separation of Variables
The given differential equation is:
x (dy/dx) = 6y
To solve this differential equation by the method of separation of variables, we need to rearrange the equation such that all the terms involving y are on one side and all the terms involving x are on the other side.
Step 1: Rearrange the Equation
We start by rearranging the equation:
dy/dx = (6y)/x
Step 2: Separate the Variables
Now, we can separate the variables y and x:
(1/y) dy = (6/x) dx
Step 3: Integrate Both Sides
Next, we will integrate both sides. The left side will be integrated with respect to y, and the right side will be integrated with respect to x:
∫(1/y) dy = ∫(6/x) dx
This gives us:
ln|y| = 6ln|x| + C
Step 4: Solve for y
To solve for y, we exponentiate both sides to eliminate the natural logarithm:
y = e^{C} |x|^{6}
Let A = e^{C}, a constant, we can rewrite the equation as:
y = A x^{6}
Conclusion
Thus, the solution to the differential equation x dy/dx = 6y is:
y = A x^{6}
where A is a constant determined by any initial conditions provided.