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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 \(

User Baju
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Final answer:

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.

User Younes
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