Final answer:
To solve the differential equation using a power series, we assume the solution as a series, find the coefficients using a recurrence relation from the original equation, and verify dimensional consistency. The process leads to the series solution of the differential equation centered at the given point x₀ = 1.
Step-by-step explanation:
To solve the differential equation y''' - xy = 0 using power series methods at the point x₀ = 1, we assume a solution in the form of a power series:
y(x) = ∑ a_n (x - x₀)^n = a_0 + a_1(x - 1) + a_2(x - 1)^2 + …
Then we find the derivatives of y and plug them into the differential equation, matching the coefficients of like powers of (x - x₀). This will give us a recurrence relation which can be used to find the coefficients a_n. Finally, we substitute back to get the series solution of the differential equation.
It is important to ensure that the equation remains dimensionally consistent; this means that all terms in the series should have the same dimensions. To verify this, if we let the dimensions of x be represented as [x] = L^aM^bT^c, we want the dimensions of each parameter to be the same for the equation to make sense.
The concept of dimensionality and power series are often summarized by the principle that you cannot add different dimensions, akin to the saying 'You can't add apples and oranges.' This principle helps with the understanding of dimensional consistency in power series expansions for various mathematical functions.