Please help with this question. I have a general solution to a differential equation of 18y^3/2= TOP LINE OF FRACTION is 2(x^2-6x+23)^3/2 and this is divided by 3. What is the particular solution for which y=2 when x=1. Then I need this particular solution in explicit form. Many thanks for help (with details). Best wishes.
Where is constant C ?
To find the particular solution for the given differential equation and initial condition, we will follow these steps:
Step 1: Rewrite the given equation in a standard form.
Step 2: Integrate both sides of the equation to find the general solution.
Step 3: Use the initial condition to find the particular solution.
Step 4: Express the particular solution in explicit form.
Let's begin with Step 1:
The given differential equation is 18y^(3/2) = (2(x^2-6x+23)^(3/2))/3.
To simplify, we can multiply both sides of the equation by 3 to eliminate the fraction:
54y^(3/2) = 2(x^2-6x+23)^(3/2).
Step 2: Integrate both sides of the equation:
∫(54y^(3/2)) dy = ∫(2(x^2-6x+23)^(3/2)) dx,
Using the power rule for integration, we have:
36/5 * y^(5/2) = 2/5 * (x^2 - 6x + 23)^(5/2) + C,
where C is the constant of integration.
Now we have the general solution of the differential equation.
Step 3: Use the initial condition y = 2 when x = 1.
Substituting x = 1 and y = 2 into the general solution,
(36/5) * 2^(5/2) = (2/5) * (1^2 - 6*1 + 23)^(5/2) + C,
72/5 = (2/5) * 18^(5/2) + C,
Simplifying,
C = 72/5 - (2/5) * 18^(5/2),
C = 72/5 - (2/5) * 18^(5/2).
Step 4: Express the particular solution in explicit form.
Substituting the value of C from Step 3 into the general solution:
36/5 * y^(5/2) = 2/5 * (x^2 - 6x + 23)^(5/2) + 72/5 - (2/5) * 18^(5/2).
Now, to express the particular solution in explicit form, we can solve for y:
y^(5/2) = [(x^2 - 6x + 23)^(5/2) + 2 * 18^(5/2)] / 36.
Taking the 2/5 * 18^(5/2) term to the other side,
y^(5/2) - [(x^2 - 6x + 23)^(5/2)]/18 = 2/5 * 18^(5/2) / 36.
Simplifying further,
y^(5/2) - [(x^2 - 6x + 23)^(5/2)]/18 = 1/5 * 18^(3/2).
Taking the 5th root on both sides,
y = [1/5 * 18^(3/2)]^(2/5) + [(x^2 - 6x + 23)^(5/2)]/18^(2/5).
Hence, the particular solution in explicit form is:
y = [1/5^(2/5) * 18^(3/5)] + [(x^2 - 6x + 23)^(5/2)]/18^(2/5).
I hope this detailed explanation helps! Let me know if you have any other questions.