Final answer:
The student seeks to solve the differential equation t²x'' - tx' + x = 0 to find solutions of the form tr and determine if these solutions form a fundamental set. By assuming a solution of the form tr and using the quadratic formula with the coefficients a = 1, b = 10, c = -2000, one can find the roots r. These roots correspond to the exponents in the assumed solution form.
Step-by-step explanation:
To find all solutions of the form tr for the differential equation t²x'' - tx' + x = 0, we assume a solution of the form x(t) = tr. Plugging into the equation and solving the characteristic equation will yield the roots (values of r), which are the exponents in the solution tr.
In order to check if the solutions form a fundamental set, they must be linearly independent and span the solution space for the differential equation. Typically for second-order linear homogeneous differential equations, if we find two solutions corresponding to different values of r, they form a fundamental set of solutions.
Solving the Quadratic Equation
To solve for t, we use the quadratic formula. We start by rearranging the quadratic equation to be t² + 10t - 2000 = 0:
√(-b ± √(b² - 4ac))/(2a)
The coefficients a, b, and c represent the respective terms in the equation where a = 1, b = 10, and c = -2000. We plug these values into the quadratic formula and calculate the solutions for t.