This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Separate the variables. The given differential equation is: x(1-y)(dy)/(dx) = -2y Divide both sides by xy: (1-y)/(y) (dy)/(dx) = -(2)/(x) Separate the variables y and x: ((1)/(y) - 1) dy = -(2)/(x) dx Step 2: Integrate both sides. Integrate both sides of the equation: ((1)/(y) - 1) dy = -(2)/(x) dx |y| - y = -2|x| + C where C is the constant of integration. Step 3: Use the initial condition to find the constant of integration. Given that y=2 when x=e: |2| - 2 = -2|e| + C Since e = 1: 2 - 2 = -2(1) + C 2 - 2 = -2 + C C = 2 Step 4: Substitute the value of C back into the general solution. |y| - y = -2|x| + 2 Rearrange the terms to match the required form. Move y to one side and logarithmic terms to the other: y = |y| + 2|x| - 2 Using logarithm properties, a b = (b^a), a + b = (ab), and a - b = ((a)/(b)): y = |y| + (x^2) - 2 y = ((|y|x^2)/(2)) Since the problem asks to show y = ((x^2y)/(2)), we can assume y > 0 and x ≠ 0. y = ((x^2y)/(2)) This matches the required form. The solution to the differential equation is y = ((x^2y)/(2)).