Question

Find the general solution of the given differential equation: 7(dy/dx) + 56y = 8?

Answer 1

To solve the differential equation 7(dy/dx) + 56y = 8, an integrating factor e^(8x) is used, leading to the general solution
`y = (1/56) + Ce^(-8x)`, where C is an arbitrary constant.

Step-by-step explanation:

The given differential equation is 7(dy/dx) + 56y = 8. To find the general solution, first we need to simplify the equation into a form that is easier to solve. We can divide the entire equation by 7 to obtain:

dy/dx + 8y = 8/7

This is now a first-order linear differential equation that can be solved using an integrating factor. The integrating factor is found using the expression
`e^(\int P dx)`, where P is the coefficient of y in the differential equation. In our case, P = 8, so the integrating factor is
`e^(8x)`. We multiply the entire differential equation by this integrating factor:

`e^(8x) dy/dx + 8e^(8x)y = (8/7)e^(8x)`

Now the left side is the derivative of the product (e^(8x)y), so we integrate both sides with respect to x:

`∫(e^(8x)y dx) = ∫((8/7)e^(8x)) dx + C`

Thus, we have:

`e^(8x)y = (1/7)e^(8x) + C`

Finally, we solve for y to get the general solution:

`y = (1/56) + Ce^(-8x)`

Where C is an arbitrary constant that represents the family of solutions to this differential equation.