MATH-2790 Differential Equations
HW assignment 1
Dennis Borisov
1. Determine the orders of the following ODEs:
(a) y ′ − y = y ′′
′
(b) y 2 = x
(c) yy ′ = x2
(d) cos(y ′ ) − xy = tan(x)
(e) (y ′ + xy)′ = ey
(f) y (3) + (y ′ )4 = x2 − x − 1
(g) yy ′ y ′′ = sin(x).
2. Determine which of the following ODEs are linear:
(a) xy (3) − (y ′ )4 + y = 0
(b) cos(x)y ′ + y = tan(x)
(c) y (2) − y (3) + x3 − x2 = ex
2
(d) ddru2 + du
dr + u = cos (r + u),
2
k
(e) ddtR
2 = − R2
x
(f) y+y
′ = 1
2
2
( dx )
dx
(g) ddt2x − 1 − dt3
dt + x = 0.
3. Find the values of A for which the given function solves the given ODE:
(a) 5y ′ = 2y and y = eAx
(b) 2y ′′ + 7y ′ − 4y = 0 and y = eAx
(c) x2 y ′′ − 7xy ′ + 15y = 0 and y = xA
6 −20t
(d) dy
dt + 20y = 24 and y = A − 5 e
(e) y ′′ + y = tan(x) and y = A cos(x) ln(sec(x) + tan(x))
√ Rx
√ dt
(f) 2xy ′ − y = Ax cos(x) and y = x 4 cos(t)
t
2
2 R x t2
′
−x
−x
(g) y + 2xy = 1 and y = A e
+e
e
dt
0
(h) 2xy + x2 − y y ′ = 0 and Ax2 y + y 2 = 1
1
4. Starting with the given family of functions find a solution to the given
IVP and determine the largest interval on which this solution is defined:
(a)
y′ = y − y2,
y=
y(−1) = 2;
1
.
1 + ce−x
(b)
1
y(−2) = ,
2
y ′ + 2xy 2 = 0,
y=
1
x2 + c
.
(c)
′
2
y=
1
.
x2 + c
y + 2xy = 0,
1
y
= −4,
2
(d)
d2 x
+ x = 0,
dt2
x
π 2
= 0,
dx π = 1,
dt 2
x = c1 cos(t) + c2 sin(t).
(e)
d2 x
+ x = 0,
dt2
x
π 4
=
√
2,
√
dx π = 2 2,
dt 4
x = c1 cos(t) + c2 sin(t).
(f)
y ′′ − y = 0,
y(1) = 0,
y = c1 ex + c2 e−x .
2
y ′ (1) = e,
(g)
y ′′ − y = 0,
y(0) = 0,
y = c1 ex + c2 e−x .
3
y ′ (0) = 0,