Find a solution of the form y1(t) = eat of the equation(3t − 1)y′′ − 9ty′ + 9y = 0,and then use the reduction of order method to find the second linearlyindependent solution.

Sure, let's solve this step by step.

Step 1: Find a solution of the form y1(t) = eat

We are given the differential equation (3t − 1)y′′ − 9ty′ + 9y = 0. We are looking for a solution of the form y1(t) = eat.

We can start by finding the derivatives of y1(t):

y1'(t) = a * e^at y1''(t) = a^2 * e^at

Substitute y1(t), y1'(t), and y1''(t) into the differential equation:

(3t - 1) * a^2 * e^at - 9t * a * e^at + 9 * e^at = 0

We can factor out e^at (since e^at ≠ 0):

e^at * [(3t - 1) * a^2 - 9t * a + 9] = 0

Setting the bracketed term equal to zero gives us a quadratic equation in a:

(3t - 1) * a^2 - 9t * a + 9 = 0

Solving this quadratic equation will give us the value of a.

Step 2: Use the reduction of order method to find the second linearly independent solution

The reduction of order method states that if y1(t) is a solution to the differential equation, then the second solution can be found by y2(t) = y1(t) * ∫(e^(-∫p(t) dt) / y1(t)^2) dt, where p(t) is the coefficient of y' in the original differential equation divided by the coefficient of y''.

In this case, p(t) = -9t / (3t - 1).

So, y2(t) = y1(t) * ∫(e^(-∫-9t / (3t - 1) dt) / y1(t)^2) dt

Solving this integral will give us the second solution y2(t).

Sure, let's solve this step by step.

Step 1: Find a solution of the form y1(t) = e^(at)

The given differential equation is (3t - 1)y'' - 9ty' + 9y = 0. We are looking for a solution of the form y1(t) = e^(at).

To find 'a', we substitute y1(t) into the differential equation and solve for 'a'.

y1'(t) = a * e^(at) y1''(t) = a^2 * e^(at)

Substituting these into the differential equation gives:

(3t - 1)a^2 * e^(at) - 9ta * e^(at) + 9e^(at) = 0

Simplifying, we get:

e^(at) * [(3t - 1)a^2 - 9ta + 9] = 0

Since e^(at) is never zero, we can divide both sides by e^(at) to get:

(3t - 1)a^2 - 9ta + 9 = 0

This is a quadratic equation in 'a'. Solving for 'a' gives a = 3t or a = -3.

Step 2: Use the reduction of order method to find the second linearly independent solution.

The reduction of order method states that if y1(t) is a solution to the differential equation, then the second solution can be found by y2(t) = y1(t) * v(t), where v(t) satisfies v'(t) = - (y1''(t) + p(t)y1'(t))/y1(t)^2.

Here, y1(t) = e^(3t) or e^(-3t), and p(t) = -9t/(3t - 1).

Substituting these into the equation for v'(t) and solving gives v(t) = e^(3t) or e^(-3t).

Therefore, the second linearly independent solution is y2(t) = y1(t) * v(t) = e^(6t) or e^(-6t).