20xy' - ln(x^5) = 0
y' = ln(x^5)/(20x) = 1/4 lnx/x
y = 1/8 (lnx)^2 + C
since y(1) = 13
y = 1/8 (lnx)^2 + 13
Differential Equation: 20xy' - ln(x^5) = 0, x>0
Initial condition: y(1) = 13
y' = ln(x^5)/(20x) = 1/4 lnx/x
y = 1/8 (lnx)^2 + C
since y(1) = 13
y = 1/8 (lnx)^2 + 13
First, let's rearrange the equation in the form of y' on one side and everything else on the other side:
20xy' = ln(x^5)
Next, divide both sides of the equation by 20x:
y' = (ln(x^5))/(20x)
Now, we can integrate both sides with respect to y and x:
∫1 dy = ∫(ln(x^5))/(20x) dx
Integrating the left side gives us:
y = C + ∫(ln(x^5))/(20x) dx
To integrate the right side, we can use u-substitution. Let u = ln(x^5), then du = (5/x) dx. We can rewrite the integral as:
∫(1/20) du
Integrating this gives us:
u/20 + C'
Now, substituting back for u, we get:
ln(x^5)/20 + C'
where C' is the constant of integration.
To find the particular solution, we can use the initial condition y(1) = 13. Plugging in x = 1 and y = 13 into the equation, we get:
13 = ln(1^5)/20 + C'
Since ln(1^5) = ln(1) = 0, the equation simplifies to:
13 = C'
Therefore, the particular solution that satisfies the initial condition y(1) = 13 is:
y = ln(x^5)/20 + 13
Step 1: Solve the differential equation to find the general solution.
Given differential equation: 20xy' - ln(x^5) = 0
To solve for y', we rearrange the equation:
20xy' = ln(x^5)
Divide both sides by 20x:
y' = ln(x^5) / (20x)
Now, integrating both sides with respect to x will give us the general solution:
∫(1/y') dy = ∫[ln(x^5) / (20x)] dx
The integral of 1/y' with respect to y is simply y because the derivative of y with respect to y is 1.
Similarly, the integral of ln(x^5) / (20x) with respect to x involves multiple steps. We can simplify it as follows:
Using the property of logarithms, ln(a^b) = b * ln(a), we have:
∫ln(x^5) / (20x) dx = ∫(5ln(x)) / (20x) dx
= (1/4) ∫(ln(x)) / x dx
Using u-substitution, let u = ln(x), then du = (1/x) dx:
(1/4) ∫(ln(x)) / x dx = (1/4) ∫u du
= (1/4)(u^2/2) + C
= (1/8)(ln^2(x)) + C
Thus, the general solution to the differential equation is given by:
y = (1/8)(ln^2(x)) + C
Step 2: Apply the initial condition to find the particular solution.
Given initial condition: y(1) = 13
Substituting x = 1 and y = 13 into the general solution, we get:
13 = (1/8)(ln^2(1)) + C
13 = 0 + C
C = 13
Therefore, the particular solution of the differential equation that satisfies the initial condition is:
y = (1/8)(ln^2(x)) + 13
This equation represents the solution to the given differential equation with the initial condition y(1) = 13.