Question

Question 1 a) Write the IVP y(4) + 3y" - sin(t)y' + 8y = t²; y(0) = 1, y'(0) = 2,y" (0) = 3, y""(0) = 4, as a system of first-order linear differential equations. b) Find the total derivative, the potential function and the solution y to the differential equation y' = (3/t) y +t⁵, t > 0. Question 2 Given t²y" - 2ty' + 2y = t³ sint, find a) the homogeneous solution, b) the general solution via the method of variation of parameters. Question 3 a) Solve the IVP y"+y' - 6y=0; y(0) = 10, y'(0) = 0. Find the homogeneous solution to y" + 4y = f(t). For each f(t), write down the particular solution y, that will solve the problem; i. f(t) = 4t² + 1, ii. f(t) = 3 sin(2t), iii. f(t) = e^2t -2t Question 4 Show that y₁=is a known solution of -t²y" + 3ty' + 5y = 0, t> 0, and find the second solution.​

Answer 1

This is a linear homogeneous second-order differential equation. We can solve it by assuming a solution of the form y = e^(rt), where r is a constant.

Plugging this into the differential equation, we get:

r²t²e^(rt) - 2rte^(rt) + 2e^(rt) = 0

Factoring out e^(rt), we get:

e^(rt)(r²t² - 2rt + 2) = 0

For a non-trivial solution, we set the expression in parentheses equal to zero:

r²t² - 2rt + 2 = 0

This is a quadratic equation in r. Solving it will give us the values of r, which will determine the homogeneous solution.

b) To find the general solution using the method of variation of parameters, we assume a particular solution of the form y = u₁(t)y₁(t) + u₂(t)y₂(t), where y₁(t) and y₂(t) are linearly independent solutions of the homogeneous equation, and u₁(t) and u₂(t) are functions to be determined.

We then find the derivatives y' and y" and substitute them into the original differential equation. This will give us a system of equations involving u₁'(t) and u₂'(t). Solving this system will give us the values of u₁'(t) and u₂'(t), which we can integrate to find u₁(t) and u₂(t). Finally, we substitute these values back into the particular solution to