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.