MAT 274 HW 1
c
Bin
Cheng
MAT 274 HW 1 Answers and Solutions
Due 11:59pm, M 8/29, 2011.
Total 80 Points since Problem 5 is postponed to HW2
1. (10’) Find the roots of
3x2 − 2x + 3 = 0
and express them using imaginary unit i.
Solution.
p
√
√
−(−2) ± (−2)2 − 4 · 3 · 3
2 ± −32
2 ± 32 i
=
=
x1,2 =
2·3
6
6
2. (10’ each) Integrate
Z
p
a)
(2y + 1) 2y − 1 dy
Solution. Integrate by substitution. Let u = 2y − 1 so that y = 12 (u + 1) and thus
2y + 1 = u + 2
Also the differential du = 2dy i.e. dy = 12 du. Plug them into the original integral
Z
Z
p
√ 1
(2y + 1) 2y − 1 dy = (u + 2) u du
2
Z
1 3/2
=
u + u1/2 du
2
1 u5/2 u3/2
=
+
+C
2 5/2
3/2
=
Z
b)
(2y − 1)5/2 2(2y − 1)3/2
+
+C
5
3
6z cos (3z 2 )dz
Solution. Integrate by substitution. Let u = 3z 2 so that du = 6z dz and the original
integeral becomes
Z
Z
2
6z cos (3z ) dz = cos u du = sin u + C = sin(3z 2 ) + C
P age
1
c
Bin
Cheng
MAT 274 HW 1
Z
c)
1
dx
x ln x2
Solution. Integrate by substitution. Let u = ln
du =
1
2
x
2
x
so that
2
1
( )0 dx = − dx
x
x
and the original integeral becomes
Z
Z
1
2 1
+C
− du = − ln |u| + C = − ln ln
2 dx =
u
x x ln x
3. (10’ each) Find general solutions (implicit if necessary, explicit if convenient) of the
following equations
a) y 0 = y 2 /(2x + 1);
Solution. Rewrite
dy
y2
=
dx
2x + 1
This is separable. Multiply with dx and divide by y 2
1
1
dy =
dx
2
y
2x + 1
Integrate
Z
1
1
dy =
dx
2
y
2x + 1
1
1
− = ln |2x + 1| + C
y
2
We can solve for y from the equation above to get an explicit form
Z
1
2 ln |2x + 1| + C
y =−1
b)
dy
dx
= ey+1 sin x.
Solution. This is separable. Multiply with dx and divide by ey+1
e−y−1 dy = sin x dx
Integrate
Z
e−y−1 dy =
Z
sin x dx
−e−y−1 = − cos x + C
i.e.
y = −1 − ln(cos x − C)
P age
2
c
Bin
Cheng
MAT 274 HW 1
4. (10’ each) Find particular solutions of
a) xy 0 − 3y = x4 ,
y(2) = 16;
Solution. Divide equation by x
dy
3
− y = x3
dx x
...(1)
Integrating factor
− x3 dx
R
ρ(x) = e
= e−3 ln(x) = x−3
Multiply ρ on equation (1)
x−3
3
dy
− x−3 y = 1
dx
x
The LHS has to be a perfect derivative
d
dx (ρy),
i.e.
d −3
(x y) = 1
dx
Integrate both sides
x−3 y = x + C
which implies
y(x) = x4 + Cx3
Plug y(2) = 16 into the above general solution
16 = 24 + C23
which yields C = 0. So the specific solution is
y(x) = x4
2
b) y 0 = e(x ) − 3y + 2xy,
y(0) = 1.
Solution. Move the y terms from RHS to LHS
dy
2
+ (3 − 2x)y = e(x )
dx
...(2)
Integrating factor
R
ρ(x) = e
3−2x dx
= e3x−x
2
Multiply ρ on equation (1)
e3x−x
2
dy
2
2
2
− e3x−x (3 − 2x)y = e3x−x e(x )
dx
P age
3
MAT 274 HW 1
The LHS has to be a perfect derivative
d
dx (ρy)
c
Bin
Cheng
and the RHS simplifies to e3x . Thus
d 3x−x2
(e
y) = e3x
dx
Integrate both sides
1
2
e3x−x y = e3x + C
3
which implies
1 2
2
y(x) = e(x ) + Cex −3x
3
Plug y(0) = 1 into the above general solution
1
1 = e0 + Ce0
3
which yields C = 23 . So the specific solution is
1 2
2 2
y(x) = e(x ) + ex −3x
3
3
5. Postponed to HW2.(10’ each) Suppose a population P is a function of time t. The
birth rate is 2 times the square of P and the death rate is a constant 8.
a) What is the differential equation that models the population dynamics? Is the equation linear or nonlinear? How many initial conditions is needed to determine a
specific solution?
b) Without help of computers or calculators, sketch the slope field and several typical
solutions of the differential obtained from previous problem. In particualr, plot 3
curves; one satisfying P (0) = 1, one satisfying P (0) = 2 and one satisfying P (0) = 4.
Is the population increasing or decreasing according to these 3 curves? (The skill of
sketching slope fields by hand is required in this class).
P age
4