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Beyond Advanced · Differential Equations

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1
The family y=ax+1/ay=ax+1/a, a≠0a\ne0, has two members through (1,3)(1,3). At that point let the normals to the two members meet the xx-axis at abscissae uu and vv. Find u2+v2u^2+v^2.
Not attempted
Two family members through Pxy-12-16P (1,3)
Your answer: —
Correct answer: 83
2
For the one-parameter family y=ax2+x/ay=ax^2+x/a, a≠0a\ne0, consider its members passing through (1,3)(1,3). If m1,m2m_1,m_2 are their tangent slopes there, which pair gives (degree of the first-order polynomial DE obtained by eliminating aa, m1m2m_1m_2)?
Not attempted
Two parameter branchesxy-12-16P (1,3)
A
(2,19)(2,19)
B
(1,19)(1,19)
C
(2,17)(2,17)
D
(2,9)(2,9)
3
For a>0a>0, let F\mathcal F be the family y=ax+1/ay=ax+1/a on x>0x>0. Select all true statements about its DE and the curve E:y=2xE:y=2\sqrt{x}.
Not attempted
Lines and the curve Exy0406E
A
At every point of EE exactly two distinct members of F\mathcal F pass through the point.
B
The member tangent to EE at abscissa tt has parameter 1/t1/\sqrt t.
C
The eliminated equation has order 11 and degree 22.
D
EE satisfies the eliminated DE but is not a member of F\mathcal F.
4
A circle has centre (a,0)(a,0) and passes through the origin, where a>0a>0. Its upper arc passes through (1,3)(1,\sqrt3). Let its tangent and normal there meet the xx-axis at TT and NN. Find TN2TN^2.
Not attempted
Circle and the given pointxy-15-33P (1,√3)centre
Your answer: —
Correct answer: 16
5
A local positive graph belongs to (x−a)2+y2=9(x-a)^2+y^2=9, with aa the sole arbitrary constant. At (0,5)(0,\sqrt5) retain only members with positive tangent slope. If p=y′p=y\prime, which equation together with which slope selects the correct local member?
Not attempted
Positive arcs through Pxy-5504P (0,√5)
A
y2(1−p2)=9y^2(1-p^2)=9, p=2/5p=2/\sqrt5
B
y2(1+p2)=9y^2(1+p^2)=9, p=2/5p=2/\sqrt5
C
y2(1+p2)=9y^2(1+p^2)=9, p=−2/5p=-2/\sqrt5
D
y(1+p2)=9y(1+p^2)=9, p=2/5p=2/\sqrt5
6
Let yy be a real C1C^1 solution on R\mathbb R of (y′)2=4y(y\prime)^2=4y, with y(0)=0y(0)=0. Select all true statements.
Not attempted
A
Every such solution is strictly positive for all x>0x>0.
B
y=x2y=x^2 is one such solution.
C
A solution may vanish on [0,5][0,5] and be positive for every x>5x>5.
D
There are infinitely many such solutions.
7
For c>0c>0, consider y=xln⁡(cx)y=x\ln(cx) on x>0x>0. A member intersects y=x2y=x^2 tangentially in the first quadrant. If p=y′p=y\prime, find the eliminated first-order equation and the value of ln⁡c\ln c.
Not attempted
Logarithmic family and y = x²xy02-15
A
xp−y=xxp-y=x, ln⁡c=0\ln c=0
B
xp+y=xxp+y=x, ln⁡c=1\ln c=1
C
xp−y=xxp-y=x, ln⁡c=1\ln c=1
D
xp−y=0xp-y=0, ln⁡c=1\ln c=1
8
A differentiable function on x>0x>0 belongs to y=(x+c)2/xy=(x+c)^2/x, where cc is constant. It satisfies y(1)=4y(1)=4 and y′(1)<0y\prime(1)<0. Let A=∫14y(x) dxA=\int_1^4 y(x)\,dx. Find (A+21/2)/ln⁡4(A+21/2)/\ln4.
Not attempted
Value condition and two branchesxy0.254010P (1,4)
Your answer: —
Correct answer: 9
9
For c>0c>0, consider the positive branch of y2=2cx−c2y^2=2cx-c^2. Eliminate cc, writing p=y′p=y\prime. Select all true statements on intervals with y>0y>0.
Not attempted
Parabolic members and y = xxy0404P (2,√3)
A
The eliminated equation has degree 22.
B
y=xy=x on x>0x>0 satisfies the eliminated DE.
C
At (2,3)(2,\sqrt3) the two allowed slopes have product 11.
D
Every positive solution of the eliminated DE is a fixed-cc member of the original family.
10
For c∈Rc\in\mathbb R, let y=x2+cy=\sqrt{x^2+c} on each positive branch. Elimination followed by differentiation-free simplification gives yy′=xyy\prime=x. A curve satisfying this DE passes through (1,2)(1,2). On its maximal interval as a positive graph, how many tangents pass through the origin?
Not attempted
Positive square-root familyxy-3305P (1,2)
A
Infinitely many
B
11
C
22
D
00
11
Let y′=(1+y2)/(1+x2)y\prime=(1+y^2)/(1+x^2), y(0)=2y(0)=2, on its maximal real interval containing 00. For how many real xx on that interval does y(x)=2xy(x)=2x?
Not attempted
A
00
B
11
C
22
D
33
12
For a>0a>0, solve y′=2xy2y\prime=2xy^2, y(0)=ay(0)=a, on its maximal interval IaI_a containing 00. Select all true statements.
Not attempted
A
For every a>0a>0, the solution extends finitely to all real xx.
B
IaI_a has length 2/a2/\sqrt a.
C
If 1∈Ia1\in I_a and y(1)=2y(1)=2, then a=2/3a=2/3.
D
For 0<a<10<a<1, ∫−11y′/y dx=0\int_{-1}^1 y\prime/y\,dx=0.
13
Let y′=2x y(1−y)y\prime=2x\,y(1-y) and y(0)=1/3y(0)=1/3. If R=∫0ln⁡22x y(x) dxR=\int_0^{\sqrt{\ln2}}2x\,y(x)\,dx, find 3eR3e^R.
Not attempted
Your answer: —
Correct answer: 4
14
A real C1C^1 function satisfies y′=2yy\prime=2\sqrt y on [0,4][0,4], y(0)=0y(0)=0 and y(4)=9y(4)=9. How many such functions exist?
Not attempted
A
00
B
11
C
22
D
Infinitely many
15
Let y′=((x+2y)2−2)/2y\prime=((x+2y)^2-2)/2, y(0)=0y(0)=0, on R\mathbb R. Find exp⁡ ⁣(2∫0ln⁡2(x+2y(x)) dx)\exp\!\left(2\int_0^{\ln2}(x+2y(x))\,dx\right).
Not attempted
Your answer: —
Correct answer: 0.64
16
Let y′=sin⁡x (y2−1)y\prime=\sin x\,(y^2-1), y(0)=0y(0)=0. Select all true statements about its real solution.
Not attempted
A
y(π)=(1−e4)/(1+e4)y(\pi)=(1-e^4)/(1+e^4).
B
The maximal interval is R\mathbb R.
C
The solution is even.
D
y(2π)=0y(2\pi)=0.
17
For a real parameter aa, the initial value problem y′=2(x−a)e−yy\prime=2(x-a)e^{-y}, y(0)=0y(0)=0, is required to have a finite real solution for every x∈Rx\in\mathbb R. For which aa is this possible?
Not attempted
A
a<1a<1
B
−1≤a≤1-1\le a\le1
C
−1<a<1-1<a<1
D
a≠±1a\ne\pm1
18
Let y′=(1+y2)/(1+x)2y\prime=(1+y^2)/(1+x)^2, y(0)=0y(0)=0. Its maximal interval containing 00 is (α,∞)(\alpha,\infty). Find 1/(1+α)1/(1+\alpha).
Not attempted
Your answer: —
Correct answer: 2.57
19
Let y′=x/(y+y3)y\prime=x/(y+y^3), y(0)=1y(0)=1, on its maximal interval containing 00. Select all true statements.
Not attempted
A
lim⁡x→∞y2/x=2\lim_{x\to\infty}y^2/x=\sqrt2.
B
yy remains positive for all real xx.
C
yy is even on R\mathbb R.
D
yy has a strict maximum at 00.
20
Let y′=(y−1)2/(1+x2)y\prime=(y-1)^2/(1+x^2), y(0)=1/2y(0)=1/2. Write LL and MM for its limits at positive and negative infinity on the initial branch. Select all true statements.
Not attempted
A
The solution has a finite left pole.
B
1/(1−L)+1/(1−M)=41/(1-L)+1/(1-M)=4
C
M>0M>0
D
L=1L=1
21
A solution of y′=ex−y−e−x−yy\prime=e^{x-y}-e^{-x-y} has y(0)=ln⁡ay(0)=\ln a, a>0a>0. For exactly two distinct real abscissae its tangent is parallel to y=3x/2y=3x/2. Determine the range of aa.
Not attempted
A
0<a<20<a<2
B
0<a≤2−25/30<a\le2-2\sqrt5/3
C
a>2+25/3a>2+2\sqrt5/3
D
0<a<2−25/30<a<2-2\sqrt5/3
22
Consider y′=y(y−1)(y−2)y\prime=y(y-1)(y-2) with y(0)=ay(0)=a, where 0<a<10<a<1. Select all true statements for the maximal initial branch.
Not attempted
A
The solution reaches 11 at a finite positive xx.
B
lim⁡x→−∞y(x)=0\lim_{x\to-\infty}y(x)=0.
C
The maximal interval is R\mathbb R.
D
The solution is strictly increasing.
23
Let xy′=y+x2/(y−x)xy\prime=y+x^2/(y-x), y(1)=2y(1)=2, on the branch with y>x>0y>x>0. At the point where y=3xy=3x, write the abscissa as ere^r. Find 2r2r.
Not attempted
Your answer: —
Correct answer: 3
24
Let y′=y/x+1+(y/x)2y\prime=y/x+1+(y/x)^2, y(1)=0y(1)=0, with x>0x>0. Select all true statements on its maximal interval.
Not attempted
A
yy has two distinct positive zeros.
B
yy is strictly increasing.
C
yy has exactly one inflection point.
D
The endpoints of the maximal interval have ratio eπe^\pi.
25
Let y>0y>0 solve 2xyy′=x2+3y22xyy\prime=x^2+3y^2, y(1)=1y(1)=1, on its maximal positive interval. What is the xx-intercept of the tangent at its unique inflection point?
Not attempted
A
4/94/9
B
2/92/9
C
2/32/3
D
−4/9-4/9
26
Let y′=(3x+y)/(x+3y)y\prime=(3x+y)/(x+3y) and y(1)=0y(1)=0. On the initial branch let PP be its first intersection, as xx increases, with y=x/2y=x/2. Find the square of the abscissa of PP.
Not attempted
Your answer: —
Correct answer: 1.92
27
Let y′=(3x+y−1)/(x+3y+5)y\prime=(3x+y-1)/(x+3y+5) and y(2)=−1y(2)=-1. At the first point on the initial branch with x>2x>2 and y+2=(x−1)/2y+2=(x-1)/2, find (x+y+1)(x−y−3)2(x+y+1)(x-y-3)^2.
Not attempted
A
00
B
The specified point does not exist.
C
11
D
8/38/3
28
Consider xy′=y+(y/x−1)(y/x+2)xxy\prime=y+(y/x-1)(y/x+2)x for x>0x>0. Select all true statements about solutions through (1,a)(1,a).
Not attempted
A
For every a>1a>1, the initial branch has a finite right endpoint.
B
y=xy=x and y=−2xy=-2x are both solutions.
C
For a=0a=0, the solution exists for every x>0x>0.
D
For a=0a=0, lim⁡x→∞y/x=1\lim_{x\to\infty}y/x=1.
29
For y>0y>0, let x=x(y)x=x(y) satisfy dx/dy=x/y+(x/y)2dx/dy=x/y+(x/y)^2, x(1)=1x(1)=1. Its positive initial branch has a vertical asymptote in the (y,x)(y,x) graph at y=b>1y=b>1. Find ln⁡b\ln b.
Not attempted
Your answer: —
Correct answer: 1
30
A positive solution of y′=(y/x)ln⁡(y/x)y\prime=(y/x)\ln(y/x) satisfies y(1)=e2y(1)=e^2. How many tangent lines to its graph on x>0x>0 pass through the origin?
Not attempted
A
22
B
11
C
00
D
Infinitely many
31
Let y′=y/x+1+(y/x)2y\prime=y/x+\sqrt{1+(y/x)^2}, y(1)=0y(1)=0, x>0x>0. Select all true statements.
Not attempted
A
The graph has exactly one positive zero.
B
The initial branch exists for every x>0x>0.
C
y′′=1y\prime\prime=1 for every x>0x>0.
D
Exactly one tangent to the graph passes through the origin.
32
Let y′=(y/x+1)/(1−y/x)y\prime=(y/x+1)/(1-y/x) and y(1)=0y(1)=0, on its maximal interval with x>0x>0. At the first point with y=x/2y=x/2, let r=x2+y2r=\sqrt{x^2+y^2}. Find ln⁡r−tan⁡−1(1/2)\ln r-\tan^{-1}(1/2).
Not attempted
Your answer: —
Correct answer: 0
33
Let y′=(2x+3y)/(3x+2y)y\prime=(2x+3y)/(3x+2y), y(1)=0y(1)=0. At the first point on the initial branch with y=x/2y=x/2, find (x+y)5/(x−y)(x+y)^5/(x-y).
Not attempted
A
4848
B
243243
C
8181
D
729729
34
For x>0x>0, consider xy′=y+xsin⁡(y/x)xy\prime=y+x\sin(y/x). A solution has y(1)=π/2y(1)=\pi/2. Select all true statements.
Not attempted
A
lim⁡x→∞y=π\lim_{x\to\infty}y=\pi.
B
y/xy/x is strictly increasing.
C
For every integer nn, y=nπxy=n\pi x solves the DE.
D
lim⁡x→0+y/x=0\lim_{x\to0^+}y/x=0.
35
Let yy solve (1+x2)y′−2xy=1+x2(1+x^2)y\prime-2xy=1+x^2 on R\mathbb R, with y(0)=ay(0)=a. For which real aa does yy have exactly one real zero?
Not attempted
A
∣a∣<π/2|a|<\pi/2
B
∣a∣≤π/2|a|\le\pi/2
C
a≠0a\ne0
D
Every real aa
36
For x≠0x\ne0, let xy′−2y=x3xy\prime-2y=x^3. Seek solutions that extend to C1C^1 functions on R\mathbb R. Select all true statements.
Not attempted
A
There are infinitely many C1C^1 extensions with y(1)=1y(1)=1.
B
Every C1C^1 extension has y′(0)=0y\prime(0)=0.
C
Under C1C^1 regularity, y(1)=1y(1)=1 uniquely determines the function on R\mathbb R.
D
A C2C^2 extension requires the two integration constants to be equal.
37
Let y′+y=2xy\prime+y=2x on [0,1][0,1], with ∫01exy(x) dx=0\int_0^1 e^x y(x)\,dx=0. Find y(1)−y(0)y(1)-y(0), correct to two decimal places.
Not attempted
Your answer: —
Correct answer: 1.09
38
A positive solution satisfies y′+xy/(1+x2)=xy3/(1+x2)y\prime+xy/(1+x^2)=xy^3/(1+x^2), y(0)=1/2y(0)=1/2. For tangents at x>0x>0, let bb be the xx-intercept. What is the minimum value of b2b^2?
Not attempted
A
16/316/3
B
32/332/3
C
8/38/3
D
44
39
Let y>0y>0 satisfy y′+y=xy2y\prime+y=xy^2, y(0)=2y(0)=2. At its unique stationary point let the abscissa be bb. Find ebe^b.
Not attempted
Your answer: —
Correct answer: 2
40
On y>0y>0, a curve is described as x=x(y)x=x(y) by y dx/dy+x=3y2y\,dx/dy+x=3y^2, with x(1)=2x(1)=2. Select all true statements.
Not attempted
Inverse curve; horizontal axis is yyx0.22.206(y,x) = (1,2)
A
The entire curve is a single-valued function yy of xx.
B
The minimum abscissa is 3/22/33/2^{2/3}.
C
The full inverse curve has exactly two positive ordinates at x=2x=2.
D
The local graph through (2,1)(2,1) has slope 11.
41
Let yy be continuous with −π/2<y<π/2-\pi/2<y<\pi/2 and satisfy sec⁡2y y′+2xtan⁡y=2x\sec^2y\,y\prime+2x\tan y=2x, y(0)=tan⁡−12y(0)=\tan^{-1}2. How many inflection points does v(x)=tan⁡y(x)v(x)=\tan y(x) have on R\mathbb R?
Not attempted
A
11
B
00
C
22
D
44
42
A positive solution on x>0x>0 satisfies y′+y/x=x2y3y\prime+y/x=x^2y^3, y(1)=1y(1)=1. Let (a,b)(a,b) be its maximal positive interval. Find bb and enter its value.
Not attempted
Your answer: —
Correct answer: 1.5
43
Let y′+2xy/(1+x2)=1/(1+x2)y\prime+2xy/(1+x^2)=1/(1+x^2) on R\mathbb R, y(0)=1y(0)=1. At its global maximum and global minimum let the values be MM and mm. Find M+mM+m.
Not attempted
A
2\sqrt2
B
22
C
00
D
11
44
Let y′+y=1+ae−xy\prime+y=1+ae^{-x}, y(0)=1y(0)=1, where a∈Ra\in\mathbb R. Select all true statements.
Not attempted
A
For a=0a=0, the solution has exactly one stationary point.
B
For a>0a>0, the global maximum equals 1+a/e1+a/e.
C
For a≠0a\ne0, the graph meets y=1y=1 at exactly one finite point.
D
For a<0a<0, the graph has a global maximum at x=1x=1.
45
Let y′+2xy=1y\prime+2xy=1, y(0)=3y(0)=3. Find e [y(1)+y(−1)]e\,[y(1)+y(-1)].
Not attempted
Your answer: —
Correct answer: 6
46
Two solutions f,gf,g of y′+P(x)y=Q(x)y\prime+P(x)y=Q(x) satisfy f(0)−g(0)=2f(0)-g(0)=2 and f(1)−g(1)=8f(1)-g(1)=8, where P,QP,Q are continuous on [0,1][0,1]. A third solution hh has h(0)=3f(0)−2g(0)h(0)=3f(0)-2g(0). Find h(1)−g(1)h(1)-g(1).
Not attempted
A
2424
B
1616
C
88
D
3232
47
A curve in the first quadrant cuts every circle x2+y2=2cxx^2+y^2=2cx orthogonally wherever they meet regularly. It passes through (1,2)(1,2). At its nonzero intersection with y=xy=x, find 8(x2+y2)8(x^2+y^2).
Not attempted
Original circle familyxy0404P (1,2)
Your answer: —
Correct answer: 25
48
The curve CC is the first-quadrant orthogonal trajectory of the family y=cx3y=cx^3 passing through (1,3)(1,3). Select all true statements about its full containing conic.
Not attempted
Original cubic familyxy0305P (1,3)
A
The first-quadrant intersection with y=xy=x is (7,7)(\sqrt7,\sqrt7).
B
The containing conic is x2+3y2=28x^2+3y^2=28.
C
The area enclosed by that conic is 28π/328\pi/\sqrt3.
D
The product of the tangent intercepts at (1,3)(1,3) is 2828.
49
The first-quadrant orthogonal trajectory of y2=cxy^2=cx through (1,2)(1,2) meets the line y=xy=x at PP. What is the area of the triangle formed by the tangent at PP and the coordinate axes?
Not attempted
Original parabolas and y = xxy0304P (1,2)
A
33
B
9/29/2
C
99
D
3/23/2
50
Let CC be the positive branch of the orthogonal trajectory of xy=cxy=c through (1,5)(1,\sqrt5). At x=2x=2, its tangent and normal meet the yy-axis at ordinates uu and vv. Find uvuv.
Not attempted
Original hyperbola familyxy0.25305x = 2P (1,√5)
A
00
B
44
C
88
D
1616
51
A first-quadrant curve satisfies (x dy−y dx)=2(x dx+y dy)(x\,dy-y\,dx)=2(x\,dx+y\,dy) and passes through (1,1)(1,1). Put r=x2+y2r=\sqrt{x^2+y^2} and θ=tan⁡−1(y/x)\theta=\tan^{-1}(y/x). Select all true statements on its connected first-quadrant arc.
Not attempted
A
The arc meets each ray y=mxy=mx, m>0m>0, exactly once.
B
The arc is part of a circle centred at the origin.
C
The limiting radii have ratio eπ/4e^{\pi/4}.
D
The radius decreases as θ\theta increases.
52
Let y>0y>0 satisfy y dx−x dy−3x2y2 dx=0y\,dx-x\,dy-3x^2y^2\,dx=0, y(1)=1/2y(1)=1/2. At its unique stationary point for x>0x>0, find x3x^3.
Not attempted
Your answer: —
Correct answer: 0.5
53
Let exy′−yex=e2x(2x+1)e^x y\prime-ye^x=e^{2x}(2x+1) and y(0)=1y(0)=1. At every stationary point of u(x)=e−xy(x)u(x)=e^{-x}y(x), consider the value of yy. What is the sum of those values?
Not attempted
A
e/4\sqrt e/4
B
3/43/4
C
1/e1/\sqrt e
D
3/(4e)3/(4\sqrt e)
54
A positive curve through (1,1)(1,1) satisfies (x dy+y dx)/(x2y2)=2x dx(x\,dy+y\,dx)/(x^2y^2)=2x\,dx. Determine the number of intersections with y=2xy=2x on its maximal positive initial branch.
Not attempted
A
22
B
00
C
11
D
44
55
A curve with x>0x>0 satisfies x dy−y dx=x2(1+(y/x)2) dxx\,dy-y\,dx=x^2(1+(y/x)^2)\,dx and y(1)=0y(1)=0. Select all true statements on its maximal initial branch.
Not attempted
A
The graph has exactly one zero on its initial interval.
B
The solution stays bounded at its right endpoint.
C
The maximal interval in x>0x>0 is (0,1+π/2)(0,1+\pi/2).
D
lim⁡x→0+y/x=0\lim_{x\to0^+}y/x=0.
56
A first-quadrant curve meets every member of y=cx2y=cx^2 orthogonally and passes through (2,1)(2,1). Let AA be the area enclosed by its full containing conic. Find A/(π2)A/(\pi\sqrt2).
Not attempted
Original parabolic familyxy0404P (2,1)
Your answer: —
Correct answer: 3
57
Let x>0x>0, y>0y>0 and (x dy−y dx)/(xy)=dx/x(x\,dy-y\,dx)/(xy)=dx/x. The initial curve passes through (2,3)(2,3). At which point does its tangent have yy-intercept −12-12?
Not attempted
A
(2,3)(2,3)
B
(4,12)(4,12)
C
(4,−12)(4,-12)
D
(8,48)(8,48)
58
For a>0a>0, the family y=ae−xy=ae^{-x} is crossed orthogonally by a curve through (0,2)(0,2). At the point of that trajectory with ordinate 11, find the slope of its tangent and the abscissa.
Not attempted
Original exponential familyxy-2205y = 1P (0,2)
A
1, 3/21,\ 3/2
B
−1, −3/2-1,\ -3/2
C
1, −3/21,\ -3/2
D
−1, 3/2-1,\ 3/2
59
A body obeys T′=−k(T−S)T\prime=-k(T-S) in a room of unknown constant temperature SS, with the same k>0k>0 throughout. Measurements at times 0,1,20,1,2 are 100,70,55100,70,55. Immediately at time 22 it is moved to a room at temperature 1010. What is T(4)T(4)?
Not attempted
Measurements before the room changetimeT040110room change1007055
A
65/265/2
B
175/4175/4
C
55/455/4
D
85/485/4
60
A tank initially contains 1010 litres of pure water. Brine of concentration 11 unit/litre enters at 22 litres/minute, and well-mixed liquid leaves at 11 litre/minute. Let t∗>0t_*>0 be the first time the concentration reaches 1/21/2. Find t∗2+20t∗t_*^2+20t_*.
Not attempted
Changing-volume, well-mixed tankV(0) = 10 litresamount A(t)in: 2 L/min1 unit/Lout: 1 L/minoutlet concentration = A(t) / V(t)
Your answer: —
Correct answer: 100
61
A substance AA decays into BB at rate A′=−kAA\prime=-kA, and BB is removed at rate 2kB2kB. Initially A=A0>0A=A_0>0, B=0B=0. Put u=A/A0u=A/A_0. Which graph (with its exact caption) represents B/A0B/A_0 as a function of uu for 0<u<10<u<1?
Not attempted
A
B/A0=u−u2B/A_0=u-u^2
Candidate graphuB/A₀01-0.31.1
B
B/A0=u2−uB/A_0=u^2-u
Candidate graphuB/A₀01-0.31.1
C
B/A0=u−u3B/A_0=u-u^3
Candidate graphuB/A₀01-0.31.1
D
B/A0=(1−u)2B/A_0=(1-u)^2
Candidate graphuB/A₀01-0.31.1
62
A particle starts from rest and obeys v′=k(V−v)v\prime=k(V-v), with constants k,V>0k,V>0. Its velocities at times 11 and 22 are 33 and 9/29/2. Let DD be its displacement from time 00 to 22. Find (12−D)ln⁡2(12-D)\ln2.
Not attempted
Measured velocity datatv02.506rest(1,3)(2,9/2)
Your answer: —
Correct answer: 4.5
63
A positive curve through (0,1)(0,1) has the following property: at (x,y)(x,y) its normal meets the xx-axis at abscissa 2x+y22x+y^2. Find y2y^2 on the initial branch.
Not attempted
A
(1/2)e2x+x+1/2(1/2)e^{2x}+x+1/2
B
(3/2)e2x−x−1/2(3/2)e^{2x}-x-1/2
C
(3/2)e−2x+x−1/2(3/2)e^{-2x}+x-1/2
D
e2x−xe^{2x}-x
64
A positive curve in x>0x>0 passes through (1,4)(1,4). At each point, its signed subtangent is y/y′=(x+1)/2y/y\prime=(x+1)/2. Among its tangents at x>1x>1, what is the infimum of the area of the triangle formed with the coordinate axes?
Not attempted
A
11
B
00, attained
C
00, not attained
D
22
65
A positive curve through (0,2)(0,2) has tangent xx-intercept (x2−1)/(2x)(x^2-1)/(2x) at every point with x≠0x\ne0. The curve is differentiable across 00. Which graph and exact equation describes it?
Not attempted
A
y=2+x2y=2+x^2
Candidate graphxy-22-711
B
y=2/(1+x2)y=2/(1+x^2)
Candidate graphxy-22-711
C
y=2(1−x2)y=2(1-x^2)
Candidate graphxy-22-711
D
y=2(1+x2)y=2(1+x^2)
Candidate graphxy-22-711
66
A body satisfies T′=−k[T−(S0+ct)]T\prime=-k[T-(S_0+ct)], where k>0k>0 and c>0c>0 are constants. Put D(t)=T(t)−(S0+ct)D(t)=T(t)-(S_0+ct) and assume D(0)>0D(0)>0. Select all true statements for t≥0t\ge0.
Not attempted
A
TT has exactly one minimum.
B
lim⁡t→∞D(t)=0\lim_{t\to\infty}D(t)=0.
C
At late times the body is colder than the ambient temperature by a limiting amount c/kc/k.
D
DD crosses zero exactly once.
67
A well-mixed fixed-volume tank has concentration C(t)C(t) satisfying C′=λ(Cin−C)C\prime=\lambda(C_{\rm in}-C), λ>0\lambda>0. The inlet alternates: Cin=1C_{\rm in}=1 for time τ\tau, then 00 for time τ\tau, repeatedly; e−λτ=1/2e^{-\lambda\tau}=1/2. In the unique periodic regime, let u=Cu=C at the start of a high-inlet stage. Which graph and exact formula gives CC during that high stage as a function of z=e−λtz=e^{-\lambda t}, 1/2≤z≤11/2\le z\le1?
Not attempted
A
C=1−2z/3C=1-2z/3
Candidate graphzC0.5101
B
C=1−z/3C=1-z/3
Candidate graphzC0.5101
C
C=2z/3C=2z/3
Candidate graphzC0.5101
D
C=1−zC=1-z
Candidate graphzC0.5101
68
A population fraction satisfies p′=p(1−p)−hp\prime=p(1-p)-h, with constant h≥0h\ge0. Only solutions with 0≤p≤10\le p\le1 are physically considered until they first leave that range. Select all true statements.
Not attempted
A
For h>1/4h>1/4, every positive initial value in [0,1][0,1] reaches 00 in finite forward time.
B
For 0<h<1/40<h<1/4, there are two distinct equilibria in (0,1)(0,1).
C
For 0<h<1/40<h<1/4, every initial value strictly between the equilibria increases toward the larger one.
D
At h=1/4h=1/4, every initial value in (0,1)(0,1) increases toward 1/21/2.
69
A positive curve through (1,2)(1,2) has the property that its normal xx-intercept is twice its point abscissa minus 1/x1/x, for x>0x>0. Let mm be the minimum of y2y^2 on its initial branch. Find em−3e^{m-3}.
Not attempted
A
11
B
ee
C
e4e^4
D
e−1e^{-1}
70
A body obeys T′=−k(T−S)T\prime=-k(T-S), k>0k>0. The ambient temperature is 00 for a time aa, then 9090 for a time bb, and this cycle repeats. Suppose e−ka=1/3e^{-ka}=1/3 and e−kb=1/2e^{-kb}=1/2. In the unique periodic regime let MM and mm be the maximum and minimum body temperatures during a cycle. Find M+mM+m.
Not attempted
A repeating ambient-temperature cyclecold: duration ahot: duration brepeatS = 0S = 90timeHorizontal durations shown schematically
Your answer: —
Correct answer: 72
Compose the two affine temperature maps, imposing periodicity only after a complete cycle.
If MM is the starting temperature of the cold stage, its endpoint is m=M/3m=M/3. In the hot stage, the endpoint is 90+(m−90)/2=45+m/290+(m-90)/2=45+m/2. Periodicity gives M=45+M/6M=45+M/6, so M=54M=54 and m=18m=18. The monotone cooling and heating within the stages show these are the cycle extrema. Therefore M+m=72M+m=72. Averaging the two ambient temperatures or treating the two durations as equal would not give the periodic extrema.