Use the derivative formulas you know to explain the Mathematica output without integration.

Here's Mathematica's formula for a solution of the differential equation y'[x] = ry [x]/x with y[1] = a Clear [y, x, a, r]; DSolve [{y' [x] == (r y[x])/x, y[1] == a}, y[x], x] {{y[x] rightarrow a x^r}} Use the derivative formulas you know to explain the Mathematica output. Here's Mathematica's formula for a solution of the differential equation y'[x] = r y [x]/x^2 with y[1] = a Clearly, [y, x, a, r]; y[x_] = y [x]/. DSolve [{y' [x])/x^2, y[1] == a}, y[x], x] [[1]] a e^r-r/x Use the derivative formulas you know to explain the Mathematica output. Here's Mathematica's formula for a solution of the differential equation y'[x] = Sin[x] y[x] with y[0] = a Clearly, [y, x, a, r]; DSolve [{y' [x] == Sin [x] y[x], y[0] == a}, y[x], x] {{y [x] rightarrow a e^1 - cos (x)}} Use the derivative formulas you know to explain the Mathematica output