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
Hey — ready when you are. Step 1: Separate the variables. The given differential equation is: 4xydx + (x^2 + 1)dy = 0 Rearrange the terms to separate x and y: (x^2 + 1)dy = -4xydx Divide both sides by y(x^2 + 1): (dy)/(y) = (-4x)/(x^2 + 1)dx Step 2: Integrate both sides. (1)/(y) dy = (-4x)/(x^2 + 1) dx For the integral on the right side, let u = x^2 + 1. Then du = 2x dx. So, -4x dx = -2(2x dx) = -2du. The integral becomes: (1)/(y) dy = (-2)/(u) du Step 3: Perform the integration. |y| = -2|u| + C_1 Substitute u = x^2 + 1 back into the equation: |y| = -2|x^2 + 1| + C_1 Step 4: Simplify the expression. Using logarithm properties, -2|x^2 + 1| = |(x^2 + 1)^-2| = ((1)/((x^2 + 1)^2)). |y| = ((1)/((x^2 + 1)^2)) + C_1 Let C_1 = |C|, where C is an arbitrary constant. |y| = ((1)/((x^2 + 1)^2)) + |C| Combine the logarithmic terms: |y| = ((C)/((x^2 + 1)^2)) Step 5: Solve for y. Exponentiate both sides: |y| = (C)/((x^2 + 1)^2) Since C is an arbitrary constant, it can absorb the absolute value sign. y = (C)/((x^2 + 1)^2) The general solution is: y = (C)/((x^2 + 1)^2) Send me the next one 📸