how do you separate these variables dy/dx=4sqrt(y)lnx/x, y(e)=9
use the fact that
integral dy/dx dx = integral dy = y
find the antiderivative and plug in e and 9 to solve for the constant... i think...
dy/y^.5 = 4 ln x /x
2 y^.5 = 2 (ln x)^2
y^.5 = (ln x)^2 + C
y = (ln x)^4 + c
(e, 9) is on it
9 = 1^4 + c
c = 8
y = (ln x)^4 + 8
To separate the variables in the given equation, we need to move all the terms involving "y" to one side of the equation and all the terms involving "x" to the other side.
The given equation is dy/dx = 4√(y)lnx/x.
Step 1: Move "x" terms to the left side and "y" terms to the right side.
dy/√y = 4lnx/x dx
Step 2: Rearrange the equation to have "dy" and "dx" terms on the same side.
√y dy = 4lnx dx/x
Step 3: Integrate both sides of the equation separately.
∫√y dy = ∫4lnx dx/x
To integrate the left side, assume u = √y and therefore du = 1/(2√y) dy, and the integral becomes:
2∫du = ∫4lnx dx/x
2u = 4∫lnx dx/x
Simplifying the right side:
2u = 4ln|x| + C1
Step 4: Substitute "u" back to "y" and solve for "y".
2√y = 4ln|x| + C1
√y = 2ln|x| + C1/2
y = (2ln|x| + C1/2)^2
Step 5: Use the initial condition y(e) = 9 to find the value of the constant C1.
Substituting x = e and y = 9 into the equation:
9 = (2ln|e| + C1/2)^2
Since ln|e| = 1, we have:
9 = (2 + C1/2)^2
Taking the square root of both sides:
±3 = 2 + C1/2
For simplicity, we'll consider the positive root:
3 - 2 = C1/2
C1 = 2
Step 6: Substitute C1 = 2 back into the equation for y.
y = (2ln|x| + 2/2)^2
y = (2ln|x| + 1)^2
So, the separated equation of variables is: y = (2ln|x| + 1)^2.
To separate variables in a differential equation, you need to rearrange the equation so that all terms involving y and its derivative dy/dx are on one side of the equation, and all terms involving x are on the other side.
In this case, the differential equation is dy/dx = 4sqrt(y)ln(x)/x. To separate the variables, we can rearrange the equation by multiplying both sides by dx:
dy = 4sqrt(y)ln(x)/x * dx
Now, we can separate the variables by moving all terms involving y to one side and all terms involving x and dx to the other side:
sqrt(y)/y * dy = 4ln(x)/x * dx
Next, we divide both sides by sqrt(y) and multiply by dx:
sqrt(y)/y * dy = 4ln(x)/x * dx
Integrating both sides will allow us to find the solution to the differential equation. To integrate the left side, we can use the substitution u = sqrt(y) or u^2 = y, which gives us:
∫ (1/u) * du = ∫ 4ln(x)/x * dx
This simplifies to:
ln(u) = 4∫ ln(x)/x * dx
Now we can integrate the right side:
ln(u) = 4∫ ln(x)/x * dx = 4∫ ln(x) * (1/x) * dx
Using integration by parts with u = ln(x) and dv = (1/x) * dx, we can find the integral:
ln(u) = 4 [ln(x) * ln(x) - ∫ (1/x) * ln(x) dx]
The integral ∫ (1/x) * ln(x) dx is a standard integral and can be evaluated as:
∫ (1/x) * ln(x) dx = ln(x) * ln(x) - x + C
where C is the constant of integration.
Therefore, the full solution to the differential equation is given by:
ln(sqrt(y)) = 4 [ln(x) * ln(x) - ln(x) + C]
To find the constant C, we can use the initial condition y(e) = 9. Substituting x = e and y = 9 into the solution equation, we get:
ln(sqrt(9)) = 4 [ln(e) * ln(e) - ln(e) + C]
ln(3) = 4 [1 - 1 + C]
ln(3) = 4C
C = ln(3)/4
Thus, the final solution to the differential equation with the given initial condition is:
ln(sqrt(y)) = 4 [ln(x) * ln(x) - ln(x) + ln(3)/4]