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Solve the differential equation y'+(3y/x+7) = (x + 7)⁶ where y = 6 when x = 0 y(x) = ______.

User Quimby
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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.

User Ibaguio
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