Question

2. Find the general solution of the differential equation. Then, use the initial condition to find the corresponding particular solution. \[ x y^{\prime}+5 y=6 x, y(1)=6 \] The general solution is \(

Answer 1

The general solution of the differential equation
`xy' + 5y = 6x` is
`y = 6\ln|x| + (5)/(x)y + C`, where
`C` is the constant of integration. The corresponding particular solution, given the initial condition
`y(1) = 6`, is
`y = 6\ln|x| + (5)/(x)y - 24`.

Step-by-step explanation:

To find the general solution of the given differential equation
`xy' + 5y = 6x`, we can use the method of separation of variables. Rearranging the equation, we have:

`y' + (5)/(x)y = 6`

Multiplying both sides of the equation by
`x`, we get:

`xy' + 5y = 6x`

This is a linear first-order differential equation. We can separate the variables by moving all terms involving
`y` to one side and all terms involving
`x` to the other side:

`y' = 6 - (5)/(x)y`

Next, we can divide both sides of the equation by
`6 - (5)/(x)y` to isolate
`y'`:

`(dy)/(dx) = (6 - (5)/(x)y)/(1)`

Now, we can integrate both sides of the equation with respect to
`x`

`\int (1)/(1) dy = \int (6 - (5)/(x)y)/(x) dx`

Simplifying the integrals, we have:

`y = \int (6 - (5)/(x)y)/(x) dx`

Integrating the right side of the equation, we get:

`y = \int (6)/(x) - (5)/(x^2)y dx`

Using the power rule of integration, we can evaluate the integral:

`y = 6\ln|x| + (5)/(x)y + C`

where
`C` is the constant of integration.

This is the general solution of the differential equation
`xy' + 5y = 6x`.

To find the corresponding particular solution, we can use the given initial condition
`y(1) = 6`. Substituting
`x = 1` and
`y = 6` into the general solution, we have:

`6 = 6\ln|1| + (5)/(1)(6) + C`

Simplifying the equation, we get:

`6 = 0 + 30 + C`

Solving for
`C`, we find:

`C = -24`

Substituting
`C = -24` back into the general solution, we obtain the particular solution:

`y = 6\ln|x| + (5)/(x)y - 24`

This is the general solution and the corresponding particular solution of the given differential equation.