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. Rearrange the differential equation to group terms involving y with dy and terms involving x with dx. (y) (dy)/(dx) = 3x^2 (x^3) (y) dy = 3x^2 (x^3) dx Step 2: Integrate both sides of the equation. (y) dy = 3x^2 (x^3) dx Step 3: Evaluate the integrals. For the left side: (y) dy = (y) + C_1 For the right side, use a substitution. Let u = x^3. Then du = 3x^2 dx. (u) du = -(u) + C_2 Substitute back u = x^3: 3x^2 (x^3) dx = -(x^3) + C_2 Step 4: Combine the results and solve for y. (y) + C_1 = -(x^3) + C_2 Let C = C_2 - C_1 be the arbitrary constant. (y) = -(x^3) + C To solve for y, take the inverse sine of both sides: y = (-(x^3) + C) The solution to the differential equation is: y = (-(x^3) + C) Send me the next one 📸