Reference no: EM133129585

Question 1: Given that y₁(t) = t² is a solution of the differential equation:

t²y" - 4ty' + 6y = 0 (t > 0)

Find the general solution.

Question 2:

If the differential equation xy" + 2y' + xe^xy = 0 has y₁ and y₂ as a fundamental set of solutions and if W(y₁, y₂)(1) = 2, find the value of W(y₁, y₂). Take x > 0.

Question 3:

Let A = (x - 2y)i + 2xj.

1. Is ∫A.dr path independent? Justify your answer.

2. Calculate A.dr along the circle of center O(0,0) and radius 1.

Question 4:

Determine the point on the plane (P): x - y + z = 3 that is closest to the point A(0,1,2).

Question 5:

Consider the sequence (x_n)_n≥1 defined as x_n= Σ_k=1^k=n 1/k^k

Show that (x_n) is bounded above. Deduce that (x_n) is convergent.