Linearly Independent

Question

If y1y1 and y2y2 are linearly independent solutions of ty″+4y′+te3ty=0ty″+4y′+te3ty=0 and if W(y1,y2)(1)=3,W(y1,y2)(1)=3, find W(y1,y2)(3).W(y1,y2)(3).

Round your answer to two decimal places.

 W(y1,y2)(3)=

Answer 1

This question asks you to compute the Wronskian of solutions to a 2nd order linear differential equation at some time given the Wronskian at a starting value. The solution to this type of equation is found using Abel's formula or identity

https://en.wikipedia.org/wiki/Abel%27s_identity

for the solutions of y'' + p(t) y' + q(t) y = 0

Abel's identity relies on p(t), which here is 4/t

W(y1,y2)(3) = W(y1,y2)(1) exp(-int_1^3 4/t dt)

int_1^3 is the integral from 1 to 3.

One thing to note about this formula is that since exp(anything) is positive, the Wronskian can't change signs. It is either always postive, negative, or zero. If the solutions are linearly independent, it is not zero