Question

Find values of m so that the function y=xᵐ is a solution of the differential equation 3x²y′′+11xy′-3y=0 Input value only. If it does not exist, input none. m = ________

Answer 1

To find values of m that make the function y = xᵐ a solution of the differential equation, we find the second derivative of y, substitute it into the equation, and solve for m. The values of m that satisfy the equation are m = 0 and m = 2.

Step-by-step explanation:

To find the values of m that make the function y = xᵐ a solution of the differential equation
`3x²y′′ + 11xy′ − 3y = 0`econd derivative of y and substitute it into the differential equation. We will then solve for m.

Using the power rule for differentiation, we find that the second derivative of y = xᵐ is y′′ = m(m-1)x^(m-2). Substituting this into the differential equation gives 3x²(m(m-1)x^(m-2)) + 11x(xᵐ) − 3(xᵐ) = 0.

Simplifying further, we have 3m(m-1)xᵐ + 11xᵐ+1 − 3xᵐ = 0. We can now equate the coefficients of the terms with the same powers of x. This gives us the equation 3m(m-1) = -3. Solving this equation for m, we find that m = 0 or m = 2.