MATH 308 – 531 Homework 1 Spring 2016

advertisement
MATH 308 – 531
Homework 1
Spring 2016
Due Thursday, Jan. 27 at the beginning of class.
1. Determine which of the following equations is liear or non linear? If the equation is non
linear, circle all the non linear terms.
(a)
(b)
(c)
(d)
(e)
d2 y
dy
3 2 + 4 + 9y = 2 cos t
dt
dt
dy
t
=3−y
dt
d2 y
dy
+ y − t2 = 0
2
dt
dt
dy
ety
=1
dt
dy √ d3 y
t2 + y 3 = 5t
dt
dt
2. Determine whether the given function is a solution to the given equation.
d2 y
+ y = t2 + 2
dt2
dy
d2 y
(b) y = 2e3t − e2t , 2 − y + 3y = −2e2t
dt
dt
(a) y = sin t + t2 ,
3. Determine for which values of m the function y(t) = tm is a solution to the equation
d2 y
dy
3t2 2 + 11t − 3y = 0.
dt
dt
4. (a) Show that the function y(t) = c1 et + c2 e−2t is a solution to the equation
d2 y dy
+
− 2y = 0 for any choice of the constants c1 and c2 .
dt2
dt
(b) Determine c1 and c2 so that the initial conditions
y(1) = 1, y ′ (1) = 0
are satisfied.
5. Solve the diļ¬€erentiate equation/initial value problem.
1 − x2
dy
=
dx
y2
(b) y −1 dy + yecos x sin x dx = 0
(a)
2
(c) (x + xy 2 )dx + ex ydy = 0
√
dy
(d)
= 2 y + 1 cos x, y(π) = 0
dx
√
(e) y dx + (1 + x)dy = 0, y(0) = 1
1
Download