If 2x = y/5 + y'/5 and (x²)d²y/dx² + λxdy/dx + ky = 0, then λ + k is equal to:

Given that 2x = y/5 + y'/5
=> 10x = y + y'
Differentiating with respect to x,
10 = y' + y''
=> y'' = 10 - y'
Substituting this into the given differential equation:
(x²)d²y/dx² + λxdy/dx + ky = 0
(x²)(10 - y') + λxy' + ky = 0
10x² - x²y' + λxy' + ky = 0
Comparing this equation with the given equation, we have:
2x = y/5 + y'/5
10x = y + y'
Differentiating with respect to x:
10 = dy/dx + d²y/dx²
d²y/dx² = 10 - dy/dx
Substituting this into the given differential equation:
(x²)d²y/dx² + λxdy/dx + ky = 0
(x²)(10 - dy/dx) + λxdy/dx + ky = 0
10x² - x²(dy/dx) + λx(dy/dx) + ky = 0
10x² + (λx - x²)dy/dx + ky = 0
This is a Cauchy-Euler equation. Let's assume a solution of the form y = xⁿ.
Then dy/dx = nxⁿ⁻¹ and d²y/dx² = n(n-1)xⁿ⁻²
Substituting into the original equation:
(x²)[n(n-1)xⁿ⁻²] + λx(nxⁿ⁻¹) + kxⁿ = 0
n(n-1)xⁿ + λnxⁿ + kxⁿ = 0
xⁿ[n(n-1) + λn + k] = 0
Since xⁿ ≠ 0, we have:
n² - n + λn + k = 0
From 10x = y + y', we have:
y' = 10x - y
Substituting this into the given equation:
(x²)d²y/dx² + λx(10x - y) + ky = 0
(x²)d²y/dx² + 10λx² - λxy + ky = 0
Let's try another approach. Since the equation is a Cauchy-Euler equation, we can assume a solution of the form y = x^m.
Then y' = mx^(m-1) and y'' = m(m-1)x^(m-2).
Substituting into the given equation:
x^2[m(m-1)x^(m-2)] + λx[mx^(m-1)] + kx^m = 0
m(m-1)x^m + λmx^m + kx^m = 0
x^m[m(m-1) + λm + k] = 0
m^2 - m + λm + k = 0
From 2x = y/5 + y'/5, we get y' = 10x - y.
Substituting into the differential equation, we get:
x²(y'') + λx(10x-y) + ky = 0
x²y'' + 10λx² - λxy + ky = 0
If this is a Cauchy-Euler equation, then the indicial equation is m(m-1) + λm + k = 0.
Comparing coefficients, we have no terms in x, so 10λx² must be 0, meaning λ = 0. Then we have m(m-1) + k = 0. This gives no information about k.
Let's go back to 10x = y + y'. Then y' = 10x - y. y'' = 10 - y'. Then y'' = 10 - (10x - y) = 10 - 10x + y.
Substituting into the DE: x²(10 - 10x + y) + λx(10x - y) + ky = 0.
This doesn't seem to lead to a solution for λ and k. There must be a mistake in the problem statement or my approach.