Hi Please can you help
Find the particular solution of the differential equation from f(x)=(x^2-6x+23)^(3/2)for which y=2 when x = 1, and then give this particular solution in explicit form?
Am I correct that the following is the explicit form (not confident).
2^2=(4/81)(1^2-6(1)+23)^(3/2)+d
d = 4/3(3-2¡Ì2)
d ¡Ö 0.228764
thank you in advance for any help
regards Claire
There are certain characters we do not see with encoding Western 8859-1.
If you could specify the encoding (of characters) you use to access Internet, or use Western ISO8859-1, we would be able to see all the characters.
Also, I do not see a y' or dy/dx in the differential equation. Could you please check?
Find the explicit formula and the three terms in the sequence after the last one given.
To find the particular solution of the given differential equation, we need to integrate the expression f(x) = (x^2 - 6x + 23)^(3/2).
First, let's find the antiderivative of f(x). We can do this by using the power rule for integration, which states that the integral of x^n is (x^(n+1))/(n+1).
So, integrating f(x) with respect to x, we have:
∫(x^2 - 6x + 23)^(3/2) dx
Let u = x^2 - 6x + 23. Then, the differential du = (2x - 6) dx.
Rearranging, we have dx = du / (2x - 6).
Substituting these values into the integral, we get:
∫u^(3/2) (du / (2x - 6))
Next, substitute u back in as x^2 - 6x + 23:
∫(x^2 - 6x + 23)^(3/2) (du / (2x - 6))
Simplifying, we have:
∫u^(3/2) du / (2(x - 3))
Now, let's integrate using the power rule:
(2/3) ∫u^(3/2) du
= (2/3) * (u^(3/2 + 1))/(3/2 + 1) + C
= (2/3) * (u^(5/2))/(5/2) + C
= (4/15) * (x^2 - 6x + 23)^(5/2) + C
Now, to find the particular solution, we need to use the initial condition y = 2 when x = 1.
Plugging in these values, we have:
2 = (4/15) * ((1)^2 - 6(1) + 23)^(5/2) + C
Simplifying, we get:
2 = (4/15) * (4/81) + C
2 = 16/2025 + C
2 = 8/10125 + C
To solve for C, subtract 8/10125 from both sides:
C = 2 - 8/10125
C = (20250 - 8)/10125
C = 20242/10125
C ≈ 1.99689
Therefore, the particular solution of the differential equation in explicit form is:
y = (4/15)(x^2 - 6x + 23)^(5/2) + 1.99689