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Find the finite polynomial solution to the differential equation x d 2 f dx2 + x df dx − 4f = 0. That is, all coefficients should be evaluated up to a single constant.

User Dch
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Answer: f(x) = a₀, where a₀ is a constant

Explanation:

To find the finite polynomial solution to the given differential equation, let's assume that the solution is in the form of a power series:

f(x) = ∑[n=0 to ∞] aₙxⁿ

where aₙ are constants to be determined.

Now, let's differentiate f(x) twice and substitute it into the differential equation:

d²f/dx² = ∑[n=0 to ∞] aₙn(n-1)xⁿ⁻²

df/dx = ∑[n=0 to ∞] aₙnxⁿ⁻¹

Substituting these derivatives into the differential equation, we get:

x(∑[n=0 to ∞] aₙn(n-1)xⁿ⁻²) + x(∑[n=0 to ∞] aₙnxⁿ⁻¹) - 4(∑[n=0 to ∞] aₙxⁿ) = 0

Next, let's combine the terms with the same power of x and simplify the equation:

∑[n=0 to ∞] aₙn(n-1)xⁿ + ∑[n=0 to ∞] aₙnxⁿ - ∑[n=0 to ∞] 4aₙxⁿ = 0

Now, we can equate the coefficients of each power of x to zero to obtain a system of equations:

a₀(0(0-1) - 4) = 0 => -4a₀ = 0

a₁(1(1-1) - 4) + a₁ = 0 => -4a₁ + a₁ = 0 => -3a₁ = 0

a₂(2(2-1) - 4) + 2a₂ - 4a₁ = 0 => -2a₂ + 2a₂ - 4a₁ = 0 => -4a₁ = 0

We observe that all the coefficients except a₀ satisfy the equation -4aₙ = 0, which means they must be zero. However, since we are looking for a non-trivial polynomial solution, a₀ cannot be zero.

Therefore, the finite polynomial solution to the given differential equation is f(x) = a₀, where a₀ is a constant.

I hope this helps :)

User PLui
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