Question

Solve the following differential equation by using integrating factors. Assume x > 0. y' + 7 ln(x) = y/X

Answer 1

To solve the given differential equation y' + 7 ln(x) = y/x using integrating factors, follow these steps: recognize the form of the equation, multiply by the integrating factor, rewrite the equation, integrate both sides, and solve for y.

Step-by-step explanation:

To solve the given differential equation, y' + 7 ln(x) = y/x, by using integrating factors, we can follow these steps:

1. Recognize that the equation is in the form y' + P(x)y = Q(x), where P(x) = 7 ln(x) and Q(x) = y/x.
2. Multiply both sides of the equation by the integrating factor, which is e to the power of the integral of P(x) dx. In this case, the integrating factor is e^(7 ln(x)) = x^7.
3. After multiplying both sides by x^7, rewrite the equation in the form (x^7 y)' = x^6 y/x.
4. Simplify the equation to (x^7 y)' = x^5 y.
5. Solve for y by isolating it on one side of the equation. In this case, we get x^7 y = (1/6)x^6 y.
6. Divide both sides of the equation by x^7 to get y = (1/6)x^6.

Therefore, the solution to the given differential equation is y = (1/6)x^6.