solve the initial value problem by separation of variables 8. dy/dx=x+1/xy, x>0, y(1)=-4

Please check that you have sufficient parentheses.

The problem posted (implicitly) is
dy/dx = x+ (1/xy) which is not easily separable.

I assume you wished to post:
dy/dx = (x+1)/xy
which can be separated as:
ydy = (x+1)dx/x = (1+1/x)dx
which can be easily integrated on both sides. The constant of integration can be obtained by substitution of the initial condition y(1)=-4.

To solve the initial value problem using separation of variables, follow these steps:

Step 1: Write the given differential equation in the form dy/dx = g(x)h(y), where g(x) is a function of x and h(y) is a function of y.

From the given problem, we have dy/dx = (x + 1)/(xy).

Step 2: Separate the variables by multiplying both sides of the equation by dx and dividing both sides by h(y).

(dy/dx) * (xy) = (x + 1)

y * dy = (x + 1) * dx

Step 3: Integrate both sides of the equation with respect to their respective variables.

∫ y * dy = ∫ (x + 1) * dx

(1/2) * y^2 = (1/2) * x^2 + x + C (Note: C is the constant of integration)

Step 4: Solve for y by taking the square root of both sides of the equation.

y = ± sqrt(x^2 + 2x + 2C)

Step 5: Use the initial condition y(1) = -4 to find the value of the constant C.

Substitute x = 1 and y = -4 in the equation.

-4 = ± sqrt(1^2 + 2(1) + 2C)

-4 = ± sqrt(1 + 2 + 2C)

-4 = ± sqrt(3 + 2C)

Squaring both sides of the equation to remove the square root, we get:

16 = 3 + 2C

2C = 16 - 3

2C = 13

C = 13/2

Step 6: Substitute the value of C back into the equation found in step 4 to obtain the final solution.

y = ± sqrt(x^2 + 2x + 13/2)

So, the solution to the initial value problem is:

y = sqrt(x^2 + 2x + 13/2) or y = -sqrt(x^2 + 2x + 13/2)

To solve the initial value problem using separation of variables, we need to separate the variables x and y and then integrate both sides.

Step 1: Separate the variables
Start by rearranging the equation to have all terms involving y on one side and all terms involving x on the other side:
dy/dx - x/y = 1/x

Step 2: Integrate both sides
Integrate both sides of the equation with respect to x:
∫(1/y)dy - ∫x/x dx = ∫(1/x)dx

Step 3: Solve the integrals
The integral of (1/y)dy can be solved by taking the natural logarithm (ln) of y:
ln|y| - ∫x dx = ln|x| + C

The integral of x/x is simply 1, so we have:
ln|y| - x = ln|x| + C

Step 4: Solve for C using the initial condition
Use the initial condition y(1) = -4 to solve for the constant C. Substitute x = 1 and y = -4 into the equation:
ln|-4| - 1 = ln|1| + C

ln(4) - 1 = 0 + C
ln(4) - 1 = C

Step 5: Rewrite the equation with the final value of C
Substitute the value of C back into the equation:
ln|y| - x = ln|x| + ln(4) - 1

Step 6: Simplify the equation
Combine the logarithmic terms on the right side:
ln|y| - x = ln(4|x|) - 1

Step 7: Exponentiate both sides to remove the logarithms
Exponentiate both sides using the base e:
e^(ln|y| - x) = e^(ln(4|x|) - 1)

Since e^ln(a) = a for any positive value a, we have:
|y|e^(-x) = 4|x|

Step 8: Solve for y
To obtain the final solution for y, we need to remove the absolute value. We can consider two cases:

Case 1: y > 0
In this case, |y| = y:
ye^(-x) = 4x

Case 2: y < 0
In this case, |y| = -y:
-y e^(-x) = 4x
Multiply through by -1:
y e^(-x) = -4x

So, we have obtained the two possible solutions for y:

1. If y > 0: ye^(-x) = 4x
2. If y < 0: y e^(-x) = -4x

These are the two separate solutions to the initial value problem.