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: (dy)/(dt) = t^2(1+y) To separate the variables, divide both sides by (1+y) and multiply by dt: (dy)/(1+y) = t^2 dt Step 2: Integrate both sides of the equation. Integrate the left side with respect to y and the right side with respect to t: (1)/(1+y) dy = t^2 dt The integral of (1)/(1+y) is |1+y|. The integral of t^2 is (t^3)/(3). So, we get: |1+y| = (t^3)/(3) + C where C is the constant of integration. Step 3: Solve for y. To isolate y, exponentiate both sides using e: e^|1+y| = e^(t^3)/(3) + C |1+y| = e^(t^3)/(3) e^C Let A = e^C. Since C is an arbitrary constant, A is an arbitrary positive constant. We can remove the absolute value by allowing A to be any non-zero constant (positive or negative). 1+y = A e^(t^3)/(3) Finally, solve for y: y = A e^(t^3)/(3) - 1 The solution to the differential equation is y = A e^(t^3)/(3) - 1. Last free one today — make it count tomorrow, or type /upgrade for unlimited.