Final answer:
The function y(x)=xᵖ is a solution of the differential equation x²y′' −3xy′ −5y=0 for the values of p=3 and p=2.
Step-by-step explanation:
The given differential equation is x²y′' −3xy′ −5y=0. We need to find the values of p for which the function y(x)=xᵖ is a solution to this equation.
To solve this, we substitute y(x)=xᵖ into the differential equation and simplify. We differentiate y(x) twice and substitute the results into the equation.
By solving the resulting equation, we find that the value of p must be 3 or 2 to satisfy the differential equation. Therefore, the correct options are a. 3 and d. 2.