Solve y dy/dx = x e^(x2 - 3y2)

This is a separable differential equation. Step 1: Separate the variables. The given equation is: y (dy)/(dx) = x e^x^2 - 3y^2 We can rewrite the exponential term using the property e^a-b = e^a e^-b: y (dy)/(dx) = x e^x^2 e^-3y^2 To separate variables, multiply both sides by e^3y^2 and by dx: y e^3y^2 dy = x e^x^2 dx Step 2: Integrate both sides of the equation. y e^3y^2 dy = x e^x^2 dx Step 3: Evaluate the integral on the left side. Let u = 3y^2. Then, differentiate u with respect to y: (du)/(dy) = 6y. So, du = 6y \, dy, which means y \, dy = (1)/(6) du. Substitute u and y \, dy into the integral: y e^3y^2 dy = e^u ((1)/(6)) du = (1)/(6) e^u du = (1)/(6) e^u + C_1 Substitute back u = 3y^2: (1)/(6) e^3y^2 + C_1 Step 4: Evaluate the integral on the right side. Let v = x^2. Then, differentiate v with respect to x: (dv)/(dx) = 2x. So, dv = 2x \, dx, which means x \, dx = (1)/(2) dv. Substitute v and x \, dx into the integral: x e^x^2 dx = e^v ((1)/(2)) dv = (1)/(2) e^v dv = (1)/(2) e^v + C_2 Substitute back v = x^2: (1)/(2) e^x^2 + C_2 Step 5: Combine the results from both integrals. (1)/(6) e^3y^2 + C_1 = (1)/(2) e^x^2 + C_2 Move the constants to one side, letting C = C_2 - C_1: (1)/(6) e^3y^2 = (1)/(2) e^x^2 + C Step 6: Simplify the general solution. Multiply the entire equation by 6 to clear the fractions: e^3y^2 = 3 e^x^2 + 6C Let K = 6C be a new arbitrary constant: e^3y^2 = 3 e^x^2 + K The general solution to the differential equation is: e^3y^2 = 3 e^x^2 + K That's 4 down. 1 left today — send the next one.