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
This is a first-order linear differential equation. We will solve it using an integrating factor. Step 1: Rewrite the differential equation in the standard linear form (dy)/(dx) + P(x)y = Q(x). The given equation is: (x+2)(dy)/(dx) - 2y = (x+2)^-1 Divide the entire equation by (x+2): (dy)/(dx) - (2)/(x+2)y = (x+2)^-1x+2 (dy)/(dx) - (2)/(x+2)y = (x+2)^-2 Here, P(x) = -(2)/(x+2) and Q(x) = (x+2)^-2. Step 2: Calculate the integrating factor, (x). The integrating factor is given by (x) = e^ P(x) \, dx. First, find P(x) \, dx: -(2)/(x+2) \, dx = -2 (1)/(x+2) \, dx = -2 |x+2| = ((x+2)^-2) Now, calculate (x): (x) = e^((x+2)^-2) = (x+2)^-2 Step 3: Multiply the standard form equation by the integrating factor. Multiplying (dy)/(dx) - (2)/(x+2)y = (x+2)^-2 by (x+2)^-2: (x+2)^-2 (dy)/(dx) - 2(x+2)^-3y = (x+2)^-4 The left side of this equation is the derivative of the product (x)y: (d)/(dx) ( (x+2)^-2 y ) = (x+2)^-4 Step 4: Integrate both sides with respect to x. (d)/(dx) ( (x+2)^-2 y ) \, dx = (x+2)^-4 \, dx (x+2)^-2 y = (x+2)^-3-3 + C (x+2)^-2 y = -(1)/(3(x+2)^3) + C Step 5: Solve for y. Multiply both sides by (x+2)^2: y = (x+2)^2 ( -(1)/(3(x+2)^3) + C ) y = -((x+2)^2)/(3(x+2)^3) + C(x+2)^2 y = -(1)/(3(x+2)) + C(x+2)^2 The general solution is: y = C(x+2)^2 - (1)/(3(x+2)) 3 done, 2 left today. You're making progress.