Download First-order Equations

First-order Equations
1. Determine, for each equation,
(a) the order.
(b) whether the equation is linear.
(c) whether the equation is separable.
(d) whether the equation is homogeneous.
• y ′ = −xy
1st order. Linear and Separable.
• y ′ + xy = x3
1st order. Linear.
• y ′ yx + x2 = y 2
1st order. Homogeneous.
x
−e
• y ′ = e 3+4y
1st order. Separable.
−x
• y ′′ + cos(t)y ′ + 2t y = tet
2nd order. Linear.
2
+3y
• y ′′′ = x 2xy
3rd order.
2
• y ′′ + xy ′ = x
2nd order. Linear and Separable.
2
+4x+2
• y ′ = 3x2(y−1)
1st order. Separable.
2. Sketch the slope field and describe the long-term behavior of the solutions.
(a) y ′ = −3y + 5
(stable) equilibrium: y = 5/3
Long term behavior: limx→∞ y =
(b) y ′ =
5
3
−y
x
(stable) equilibrium: y = 0
Long term behavior: limx→∞ y = 0
1
(c) y ′ = −3y 2 + 21y − 30
(stable) equilibrium: y = 5
(unstable) equilibrium: y = 2

 −∞,
2,
Long term behavior: limx→∞ y =

5,
if y(0) < 2
if y(0) = 2
if y(0) > 2
3. Solve the IVP and determine the domain of your solution:
12 − xy ′ = 3y
(a)
y(1) = 3
xy ′ + 3y = 12
y′ +
µ(x) = e
R
x3 y = 4x3 + C
3
12
y=
x
x
3
x dx
3=4+C
C = −1
= e3 ln x = x3
y =4−
(x3 y)′ = x3 y ′ + 3x2 y = 12x2
1
x3
Domain: 0 < x < ∞.
(b)
(
2
y
y ′ = yx2 −1
y(3) = e
(y 2 − 1)dy
= x2 dx
y
Z
Z
1
(y − )dy = x2 dx
y
1
1 2
e − 1 = (33 ) + C
2
3
1 2
C = e − 10
2
1 2
1
y − ln |y| = x3 + C
2
3
1 2
2y
63 3 2
− e 6= x3
2
2
r
3 3
x 6=
(21 − e2 ) < 3
2
q
Domain: 3 32 (21 − e2 ) < x < ∞ Whew!
Domain:
y 2 − 1 6= 0
y 6= ±1
1
1
1
6= x3 + e2 − 10
2
3
2
(c)
y ′ − 4tan(4t)y = et
y(0) = 0
µ(t) = e
− ln |y| = 31 x3 + 21 e2 − 10
R
−4 tan(4t)dt
= eln(cos(4t)) = cos(4t)
(cos(4t)y)′ = cos(4t)y ′ − 4 sin(4t)y = et cos(4t)
2
cos(4t)y =
Z
et cos(4t)dt = Re
Z
e(1+4i)t dt = Re
1
e(1+4i)t dt + C
1 + 4i
1 t
4
1 − 4i (1+4i)t
e
dt + C =
e cos(4t) + et sin(4t) + C
cos(4t)y = Re
17
17
17
y=
y=
1 t
17 e
+
4 t
17 e
1 t
4
e + et tan(4t) + C
17
17
1
0=
+C
17
1
C =−
17
tan(4t) −
3
1
17
Domain: − π8 < x <
π
8