Hi Please can you help?

Find the particular solution of the differential equation from
dy/dx=2/27(x-3)�ã(x^2-6x+23) / (y) (y>0.
for which y = 2 when x = 1, and then give this particular solution in explicit form?
thank you in advance for any help
regards Claire

Of course, Claire! I'd be happy to help you with that.

To find the particular solution of the given differential equation, we can use the method of separation of variables. Here's how you can approach it step by step:

Step 1: Rewrite the differential equation in a suitable form:
dy/dx = 2/27(x - 3) * (x^2 - 6x + 23) / y

Step 2: Separate the variables:
y dy = 2/27(x - 3) * (x^2 - 6x + 23) dx

Step 3: Integrate both sides of the equation:
∫y dy = ∫(2/27(x - 3) * (x^2 - 6x + 23)) dx

Step 4: Evaluate the integrals:
∫y dy = (2/27) ∫((x - 3) * (x^2 - 6x + 23)) dx

The integral of y dy is simply (1/2) y^2, and the integral inside the parentheses can be expanded and integrated term by term.

Step 5: Simplify the integrals:
(1/2) y^2 = (2/27) ∫(x^3 - 9x^2 + 23x - 69) dx

Integrating term by term, we get:
(1/2) y^2 = (2/27) * (1/4)x^4 - (2/27) * (3/5)x^5 + (2/27) * (23/2)x^2 - (2/27) * 69x + C

Step 6: Apply the initial condition:
Substitute the given initial condition y = 2 when x = 1 into the equation.

(1/2) (2^2) = (2/27) * (1/4) - (2/27) * (3/5) + (2/27) * (23/2) - (2/27) * 69 + C

Simplify the equation and solve for C.

1 = 1/54 - 2/45 + 23/27 - 138/27 + C
1 = -4/810 + 36/810 + 810/810 - 3006/810 + C
1 = 540/810 - 3636/810 + C
1 = -3096/810 + C
1 = -34/9 + C

Adding 34/9 to both sides:
1 + 34/9 = -34/9 + C + 34/9
43/9 = C

Step 7: Substitute the value of C back into the particular solution:
(1/2) y^2 = (2/27) * (1/4)x^4 - (2/27) * (3/5)x^5 + (2/27) * (23/2)x^2 - (2/27) * 69x + 43/9

Simplifying further, we get:
(1/2) y^2 = (1/54)x^4 - (2/45)x^5 + (23/27)x^2 - (2/27) * 69x + 43/9

Finally, we can write the particular solution explicitly in the form:
y = sqrt[(108/27)x^4 - (216/45)x^5 + (46/27)x^2 - (4/27) * 69x + 86/9]

That's it! The particular solution of the given differential equation, satisfying the initial condition y = 2 when x = 1, is y = sqrt[(108/27)x^4 - (216/45)x^5 + (46/27)x^2 - (4/27) * 69x + 86/9].