due September 25, 2015
Topics covered : exact equations: the method of integrating factor, (sections 2.6);Wronskian, fundamental set of solutions of linear homogeneous equations of second order (section 3.2).
1. For the differential equation
6 xy + 5( x
2
+ y ) d y d x
= 0 find the integrating factor depending on y only to make it exact and then solve the equation.
2.
(a) Find the Wronskian of the functions y
1
( t ) = t 2 sin(ln t 3 ), y
2
( t ) = t 2 cos(ln t 3 ), t > 0;
(b) Show that functions y
1
( t ) and y
2
( t ) from the previous item are solutions of the differential equation t
2 y
00
− 3 ty
0
+ 13 y = 0 , t > 0 (1)
(c) Show that the functions y
1
( t ) and y
2
( t ) from item a) constitute the fundamental set of solutions of the equation (1) and find the general solution of the equation (1)
(d) Find the solution of (1) satisfying the initial conditions y (1) = 7, y
0
(1) = 6.
3. ( bonus-25 points ) Read the formulation and the proof of Abel’s (Theorem Theorem 3.2.7 in
10th edition and Theorem 3.2.6 in ninth edition). Then, using this theorem, find the value of
W ( y ty
00
1
, y
+ 3 y
2
0
)(2), if the functions y
1
+ sin( t ) y
( t
= 0 such that
) and
W ( y y
1
2
(
, y t
2
) constitute a fundamental set of solutions of the equation
)(5) = 2 (for this you do not actually need to find y
1
( t ) and y
2
( t )).
1