Given the differential equation start fraction, d, y, divided by, d, x, end fraction, equals, start fraction, 2, x, minus, 1, divided by, 2, y, end fraction, comma dxdy = 2y2x−1 , find the particular solution, y, equals, f, of, x, commay=f(x), with the initial condition f, of, minus, 4, equals, minus, 2, .f(−4)=−2.
Question
Given the differential equation start fraction, d, y, divided by, d, x, end fraction, equals, start fraction, 2, x, minus, 1, divided by, 2, y, end fraction, comma dxdy = 2y2x−1 , find the particular solution, y, equals, f, of, x, commay=f(x), with the initial condition f, of, minus, 4, equals, minus, 2, .f(−4)=−2.
Solution
The given differential equation is dy/dx = (2x - 1) / (2y). This is a separable differential equation, which means we can rearrange it to isolate the variables y and x on different sides of the equation.
Step 1: Rearrange the equation We can multiply both sides by dx and by 2y to get: 2y dy = (2x - 1) dx
Step 2: Integrate both sides We can now integrate both sides of the equation with respect to their respective variables: ∫2y dy = ∫(2x - 1) dx
This gives us: y^2 = x^2 - x + C
Step 3: Solve for y We can solve for y by taking the square root of both sides. However, we must remember that the square root of a square is the absolute value, so we get two solutions: y = sqrt(x^2 - x + C) and y = -sqrt(x^2 - x + C)
Step 4: Apply the initial condition We are given the initial condition f(-4) = -2. We can substitute these values into our solutions to solve for C.
For y = sqrt(x^2 - x + C), we get: -2 = sqrt((-4)^2 - (-4) + C) Solving for C, we find that no real value of C satisfies this equation, so this solution is not valid.
For y = -sqrt(x^2 - x + C), we get: -2 = -sqrt((-4)^2 - (-4) + C) Solving for C, we find that C = -12.
Step 5: Write the final solution So, the particular solution to the differential equation with the given initial condition is: y = -sqrt(x^2 - x - 12)
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