Final answer:
To solve the given differential equation using the method of integrating factors, rearrange the equation to isolate the y' term, multiply both sides by x + 7, recognize that the left side can be written as the derivative of (x + 7)y, integrate both sides, and then solve for y using the initial condition.
Step-by-step explanation:
To solve the given differential equation, we can use the method of integrating factors. First, rearrange the equation to isolate the y′ term: y′ = (x + 7)⁶ - (3y/x + 7). Then, multiply both sides of the equation by x + 7 to obtain: (x + 7)y′ = (x + 7)⁷ - 3y.
Next, recognize that the left side of the equation can be written as the derivative of (x + 7)y with respect to x: (x + 7)y′ + 7y = (x + 7)⁷.
Finally, integrate both sides of the equation to find the solution: ∫[(x + 7)y′ + 7y]dx = ∫(x + 7)⁷dx.
This gives us (x + 7)y + 7∫ydx = (1/8)(x + 7)⁸ + C, where C is the constant of integration. We can then solve for y by substituting the initial condition y = 6 when x = 0 into the equation.