A Let gt be the solution of the initial value problem A Let

A. Let g(t) be the solution of the initial value problem A. Let g(t) be the solution of the inial value problem 2tdy +y=0, dy dt t>0 with g(1) = 1. Find g(t) g(t) = B. Let f(t) be the solution of the initial value problem dt with f(0)- 0 Find f (t). f(t) C. Find a constant C so that k(t) = f(t) + cg (t) solves the differential equation in part B and k(1)=2.

Solution

Find the general solution by separating the variables and integrating:
2ty\' + y = 0
2t(dy / dt) + y = 0
2t(dy / dt) = -y
dy / y = -1 / 2t dt
1 / y dy = - 1 / t dt / 2
ln|y| = -ln|t| / 2 + C
ln|y| = C - ln|t| / 2
y = ^(C - ln|t| / 2)
y = ^(C - ln|t|)
y = / 2t
y = C / 2t

Find the particular solution by solving for the constant:
When t = 1, y = 1
C = 1
y = 1 / 2t
g(t) = 1 / 2t