Find the particular solution of the differential equation that satisfies the initial condition. Write it out as an equation
Differential Equation: 20xy' - ln(x^5) = 0, x>0
Initial condition: y(1) = 13
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
Thank you so much that really helps!
To find the particular solution of the differential equation, we can use separation of variables.
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
To find the particular solution of the differential equation that satisfies the given initial condition, we need to follow these steps:
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.