AP Calculus AB · Flashcards
Unit 5 — Differential Equations
Course framework: Unit 7 · cards on Euler’s method and the logistic model are marked BC only · 69 cards. Answer each card in your head, flip it, then mark it Got it or Review again. Your progress is saved in this browser.
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DE Basics & Slope Fields✓ Known #1
What is a differential equation?
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Answer#1
An equation that relates a function to one or more of its derivatives.
Trap: Thinking any equation containing \(y\) is a differential equation.
DE Basics & Slope Fields✓ Known #2
What is a solution to \(\dfrac{dy}{dx}=f(x,y)\)?
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Answer#2
A differentiable function \(y(x)\) that satisfies the equation for every \(x\) in an interval.
Trap: Checking a single point instead of the whole function.
DE Basics & Slope Fields✓ Known #3
How do you verify that a given function is a solution of a differential equation?
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Answer#3
- Differentiate the function.
- Substitute \(y\) and \(y’\) into the equation and check that both sides are equal for all \(x\).
Trap: Substituting only one value of \(x\).
DE Basics & Slope Fields✓ Known #4
General solution vs particular solution?
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Answer#4
- General: a family of solutions with an arbitrary constant.
- Particular: the one member that satisfies a given initial condition.
Trap: Calling the general solution the one that uses the initial condition.
DE Basics & Slope Fields✓ Known #5
What is an initial value problem (IVP)?
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Answer#5
A differential equation together with an initial condition \(y(x_0)=y_0\), which picks out one particular solution.
Trap: Leaving the constant \(C\) unsolved.
DE Basics & Slope Fields✓ Known #6
In a slope field for \(\dfrac{dy}{dx}=f(x,y)\), what does each segment represent?
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Answer#6
The slope \(\frac{dy}{dx}\) of the solution curve through that point \((x,y)\).
Trap: Thinking the segments are pieces of one solution curve.
DE Basics & Slope Fields✓ Known #7
If \(\dfrac{dy}{dx}\) depends only on \(x\), what pattern does the slope field show?
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Answer#7
All segments in the same vertical column (same \(x\)) have the same slope.
Trap: Expecting the slopes to change with \(y\).
DE Basics & Slope Fields✓ Known #8
If \(\dfrac{dy}{dx}\) depends only on \(y\), what pattern does the slope field show?
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Answer#8
All segments in the same horizontal row (same \(y\)) have the same slope.
Trap: Mixing up rows and columns.
DE Basics & Slope Fields✓ Known #9
What is an equilibrium solution of \(\dfrac{dy}{dx}=g(y)\)?
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Answer#9
A constant solution \(y=c\) with \(g(c)=0\).
Trap: Setting \(g'(c)=0\) instead.
DE Basics & Slope Fields✓ Known #10
How do you sketch a solution curve from a slope field and an initial condition?
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Answer#10
Start at the initial point and draw a smooth curve that stays tangent to the nearby segments.
Trap: Drawing a curve that cuts across the segments.
DE Basics & Slope Fields✓ Known #11
If the slopes are positive in a region, what happens to solutions there?
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Answer#11
They increase as \(x\) increases.
Trap: Confusing positive slope with concave up.
DE Basics & Slope Fields✓ Known #12
If the slopes are zero along the horizontal line \(y=c\), what is special about it?
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Answer#12
\(y=c\) is an equilibrium solution: a solution starting there stays constant.
Trap: Thinking it is just a point of tangency.
DE Basics & Slope Fields✓ Known #13
How do you find \(\dfrac{d^2y}{dx^2}\) from \(\dfrac{dy}{dx}=x+y\)?
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Answer#13
- Differentiate implicitly: \(\dfrac{d^2y}{dx^2}=1+\dfrac{dy}{dx}\).
- Substitute \(\frac{dy}{dx}\): \(\dfrac{d^2y}{dx^2}=1+x+y\). Its sign gives the concavity of the solution.
Trap: Treating \(y\) as a constant when differentiating.
DE Basics & Slope Fields✓ Known #14
How do you write the tangent line to a solution curve at \((x_0,y_0)\)?
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Answer#14
Use the differential equation for the slope: \(m=f(x_0,y_0)\), then \(y-y_0=m(x-x_0)\).
Trap: Trying to solve the differential equation first.
Separation of Variables✓ Known #15
When is a differential equation separable?
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Answer#15
When it can be written as \(\dfrac{dy}{dx}=p(x)\,q(y)\), so the \(x\)- and \(y\)-terms can be separated.
Trap: Trying to separate terms that are added rather than multiplied.
Separation of Variables✓ Known #16
Separation of variables: what are the 4 steps?
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Answer#16
- Write \(\frac{dy}{dx}=p(x)q(y)\).
- Move the \(y\)-terms to \(dy\) and the \(x\)-terms to \(dx\).
- Integrate both sides, with one constant \(C\).
- Use the initial condition to find \(C\), then solve for \(y\).
Trap: Forgetting \(+C\), or adding it only after exponentiating.
Separation of Variables✓ Known #17
Separate the variables in \(\dfrac{dy}{dx}=xy\).
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Answer#17
\(\dfrac{1}{y}\,dy=x\,dx\).
Trap: Writing \(dy=xy\,dx\) and integrating with \(y\) still on the right.
Separation of Variables✓ Known #18
Integrate both sides of \(\dfrac{1}{y}\,dy=x\,dx\).
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Answer#18
\(\ln|y|=\dfrac{x^2}{2}+C\).
Trap: Forgetting the absolute value.
Separation of Variables✓ Known #19
From \(\ln|y|=\dfrac{x^2}{2}+C\), solve for \(y\).
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Answer#19
\(y=Ae^{x^2/2}\), where \(A\) is any real number (\(A=0\) gives the equilibrium \(y=0\)).
Trap: Writing \(y=e^{x^2/2}+C\).
Separation of Variables✓ Known #20
Why can the constant absorb the \(\pm\) sign after exponentiating?
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Answer#20
\(|y|=e^{C}e^{x^2/2}\) gives \(y=\pm e^{C}e^{x^2/2}\); the factor \(\pm e^{C}\) is just some nonzero constant \(A\).
Trap: Keeping \(\pm\) and \(e^C\) separately and making algebra errors.
Separation of Variables✓ Known #21
When can you drop the absolute value in \(\ln|y|\)?
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Answer#21
When the initial condition shows \(y\) stays positive (or negative) on the interval.
Trap: Dropping it with no justification.
Separation of Variables✓ Known #22
Separate the variables in \(\dfrac{dy}{dx}=\dfrac{y}{x}\) \((x\ne0)\).
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Answer#22
\(\dfrac{1}{y}\,dy=\dfrac{1}{x}\,dx\).
Trap: Multiplying the wrong side by \(x\) or \(y\).
Separation of Variables✓ Known #23
Integrate \(\dfrac{1}{y}\,dy=\dfrac{1}{x}\,dx\) and solve for \(y\).
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Answer#23
\(\ln|y|=\ln|x|+C\Rightarrow y=Ax\).
Trap: Stopping at \(\ln y=\ln x\) and losing the constant.
Separation of Variables✓ Known #24
Separate the variables in \(\dfrac{dy}{dx}=x(y^2+1)\).
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Answer#24
\(\dfrac{1}{y^2+1}\,dy=x\,dx\).
Trap: Putting \(y^2+1\) on the wrong side.
Separation of Variables✓ Known #25
\(\displaystyle\int \frac{1}{y^2+1}\,dy=\,\)?
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Answer#25
\(\arctan y+C\).
Trap: Writing \(\ln(y^2+1)\).
Separation of Variables✓ Known #26
From \(\arctan y=\dfrac{x^2}{2}+C\), solve for \(y\).
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Answer#26
\(y=\tan\left(\dfrac{x^2}{2}+C\right)\).
Trap: Applying \(\tan\) to only one term.
Separation of Variables✓ Known #27
Given \(y(0)=2\), how do you use it with a separable equation?
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Answer#27
Substitute \(x=0,\ y=2\) right after integrating to find \(C\), then solve for \(y\).
Trap: Substituting into the differential equation instead of the solution.
Separation of Variables✓ Known #28
What can you lose when you divide by \(q(y)\) to separate?
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Answer#28
Equilibrium solutions where \(q(y)=0\) (for example \(y=0\)). Check them separately.
Trap: Missing a constant solution.
Exponential Models✓ Known #29
Exponential growth/decay differential equation?
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Answer#29
\(\dfrac{dy}{dt}=ky\): the rate is proportional to the amount.
Trap: Using \(\frac{dy}{dt}=k\), which gives linear change.
Exponential Models✓ Known #30
Solution of \(\dfrac{dy}{dt}=ky\)?
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Answer#30
\(y=y_0e^{kt}\), where \(y_0=y(0)\).
Trap: Writing \(y=Ce^{k}\).
Exponential Models✓ Known #31
If \(k>0\), what happens as \(t\) increases?
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Answer#31
Exponential growth: \(y\) increases without bound (for \(y_0>0\)).
Trap: Calling it decay.
Exponential Models✓ Known #32
If \(k<0\), what happens as \(t\) increases?
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Answer#32
Exponential decay: \(y\to0\).
Trap: Thinking \(y\) becomes negative.
Exponential Models✓ Known #33
What does half-life mean for \(y=y_0e^{kt}\) with \(k<0\)?
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Answer#33
The time for the amount to be multiplied by \(\frac12\): \(t_{1/2}=\dfrac{\ln 2}{-k}\).
Trap: Confusing it with the time to reach 0.
Exponential Models✓ Known #34
What is the doubling time for \(k>0\)?
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Answer#34
The time for the amount to double: \(t=\dfrac{\ln 2}{k}\). It does not depend on the initial amount.
Trap: Thinking it depends on \(y_0\).
Exponential Models✓ Known #35
For \(y=Ce^{kt}\), how do you find \(C\) from \(y(0)=y_0\)?
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Answer#35
\(C=y_0\).
Trap: Substituting into the derivative instead of the solution.
Exponential Models✓ Known #36
“The rate of change is proportional to the amount present.” Which equation is this?
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Answer#36
\(\dfrac{dy}{dt}=ky\).
Trap: Writing \(y’=k\).
Exponential Models✓ Known #37
If \(y(t)=y_0e^{kt}\), what is \(\dfrac{y(t_2)}{y(t_1)}\)?
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Answer#37
\(e^{k(t_2-t_1)}\).
Trap: Leaving \(y_0\) in the ratio.
Exponential Models✓ Known #38
Given \(y(t_1)\) and \(y(t_2)\), how do you find \(k\)?
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Answer#38
\(k=\dfrac{1}{t_2-t_1}\ln\!\left(\dfrac{y(t_2)}{y(t_1)}\right)\).
Trap: Using \(\log_{10}\) instead of \(\ln\).
Exponential Models✓ Known #39
Newton’s law of cooling: write the differential equation.
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Answer#39
\(\dfrac{dT}{dt}=k(T-A)\), where \(A\) is the surrounding temperature and \(k<0\). Solution: \(T=A+(T_0-A)e^{kt}\).
Trap: Writing \(\frac{dT}{dt}=kT\), which ignores the room temperature.
Logistic Model (BC)✓ Known #40
BC only · Logistic differential equation?
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Answer#40
\(\dfrac{dP}{dt}=kP\left(1-\dfrac{P}{L}\right)\), where \(L\) is the carrying capacity.
Trap: Forgetting the factor \(\left(1-\frac PL\right)\).
Logistic Model (BC)✓ Known #41
BC only · What is the carrying capacity?
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Answer#41
The limiting value \(L\) that \(P\) approaches as \(t\to\infty\) (for \(k>0\), \(P_0>0\)).
Trap: Calling it the initial population.
Logistic Model (BC)✓ Known #42
BC only · Equilibrium solutions of the logistic equation?
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Answer#42
\(P=0\) and \(P=L\).
Trap: Solving \(P’=0\) incorrectly, for example getting \(P=k\).
Logistic Model (BC)✓ Known #43
BC only · If \(0<P<L\) and \(k>0\), what is the sign of \(\frac{dP}{dt}\)?
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Answer#43
Positive: \(P\) increases.
Trap: Thinking the factor \(\left(1-\frac PL\right)\) makes it negative.
Logistic Model (BC)✓ Known #44
BC only · If \(P>L\) and \(k>0\), what happens to \(P\)?
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Answer#44
\(1-\frac PL<0\), so \(P'<0\): \(P\) decreases toward \(L\).
Trap: Thinking a large \(P\) keeps growing.
Logistic Model (BC)✓ Known #45
BC only · Where does a logistic solution grow fastest?
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Answer#45
At \(P=\dfrac{L}{2}\), which is also where the solution curve has its inflection point.
Trap: Saying it grows fastest near \(P=L\).
Logistic Model (BC)✓ Known #46
BC only · Shape of a logistic solution with \(k>0\) and \(0<P_0<\frac L2\)?
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Answer#46
S-shaped: increasing, concave up until \(P=\frac L2\), then concave down, approaching \(L\).
Trap: Calling it exponential forever.
Logistic Model (BC)✓ Known #47
BC only · Why does logistic growth slow down?
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Answer#47
The factor \(1-\frac{P}{L}\) shrinks toward 0 as \(P\) approaches \(L\).
Trap: Saying that \(k\) changes.
Logistic Model (BC)✓ Known #48
BC only · For \(\dfrac{dP}{dt}=0.2P\left(1-\dfrac{P}{500}\right)\), find \(k\), \(L\) and where growth is fastest.
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Answer#48
- \(k=0.2\), \(L=500\).
- Growth is fastest when \(P=250\).
Trap: Reading \(L\) from the coefficient instead of the denominator.
Logistic Model (BC)✓ Known #49
BC only · Interpret \(\frac{dP}{dt}\) when \(P\) is close to \(L\).
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Answer#49
It is close to 0: the population is changing slowly because it is near capacity.
Trap: Claiming the rate is large because \(P\) is large.
Logistic Model (BC)✓ Known #50
BC only · What is \(\displaystyle\lim_{t\to\infty}P(t)\) for \(k>0\) and \(P_0>0\)?
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Answer#50
\(L\), the carrying capacity.
Trap: Answering 0 or \(\infty\).
Logistic Model (BC)✓ Known #51
BC only · For \(k>0\), which equilibrium is stable?
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Answer#51
\(P=L\) is stable (nearby solutions move toward it); \(P=0\) is unstable.
Trap: Calling both stable.
Euler’s Method (BC)✓ Known #52
BC only · Euler’s method update formula?
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Answer#52
\(y_{n+1}=y_n+f(x_n,y_n)\,\Delta x\).
Trap: Using the slope at \((x_{n+1},y_{n+1})\).
Euler’s Method (BC)✓ Known #53
BC only · How does \(x\) change in each Euler step?
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Answer#53
\(x_{n+1}=x_n+\Delta x\).
Trap: Changing \(x\) by the slope.
Euler’s Method (BC)✓ Known #54
BC only · What does \(f(x_n,y_n)\) represent in Euler’s method?
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Answer#54
The tangent slope at the current point.
Trap: Using the slope from the next point.
Euler’s Method (BC)✓ Known #55
BC only · “Use Euler’s method with step size \(h\).” What is \(h\)?
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Answer#55
\(h=\Delta x\), the \(x\)-increment of each step.
Trap: Confusing \(h\) with the change in \(y\).
Euler’s Method (BC)✓ Known #56
BC only · If the slope is negative and the step is positive, what happens to \(y\) in one step?
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Answer#56
\(y\) decreases, because \(y_{n+1}=y_n+(\text{negative})(\text{positive})\).
Trap: Thinking \(y\) increases because you “add.”
Euler’s Method (BC)✓ Known #57
BC only · \(y’=x+y\), \(y(0)=1\), \(h=0.1\). Find \(y_1\).
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Answer#57
Slope at \((0,1)\): \(0+1=1\). So \(y_1=1+1(0.1)=1.1\).
Trap: Using \(x=0.1\) in the first slope.
Euler’s Method (BC)✓ Known #58
BC only · What is the geometric idea of Euler’s method?
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Answer#58
Follow the tangent line for a short step, then recompute the slope at the new point.
Trap: Thinking it gives the exact solution.
Euler’s Method (BC)✓ Known #59
BC only · What happens to Euler’s method as \(h\) gets smaller?
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Answer#59
It usually becomes more accurate (with more steps).
Trap: Assuming larger steps are better.
Euler’s Method (BC)✓ Known #60
BC only · Which columns should you track in an Euler table?
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Answer#60
\(x_n\), \(y_n\), the slope \(f(x_n,y_n)\), and \(y_{n+1}\).
Trap: Skipping the slope column.
Euler’s Method (BC)✓ Known #61
BC only · How should you handle rounding in Euler’s method?
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Answer#61
Carry 3–4 decimal places through every step and round only the final answer.
Trap: Rounding at every step.
Qualitative Behavior✓ Known #62
If \(y’=g(y)\) and \(g(y)>0\) for \(2<y<5\), what happens to solutions in that band?
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Answer#62
They increase while they stay in the band.
Trap: Using \(g'(y)\) instead of \(g(y)\).
Qualitative Behavior✓ Known #63
If \(y’=g(y)\) and \(g(y)<0\) for \(y>5\), what happens to solutions above 5?
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Answer#63
They decrease (they are pushed down toward 5).
Trap: Assuming large \(y\) keeps growing.
Qualitative Behavior✓ Known #64
How do you decide whether an equilibrium \(y=c\) of \(y’=g(y)\) is stable?
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Answer#64
Check the sign of \(g\) just above and below \(c\): if solutions move toward \(c\) from both sides, it is stable; if away, unstable.
Trap: Checking only \(g(c)=0\).
Qualitative Behavior✓ Known #65
If \(y’\) is approaching 0, what does that suggest about the solution curve?
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Answer#65
The curve is leveling off — its slope is approaching horizontal.
Trap: Concluding that \(y\) itself is approaching 0.
Common Traps✓ Known #66
Separation trap: what should you do before dividing by \(y\)?
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Answer#66
Check whether \(y=0\) is itself a solution.
Trap: Losing the equilibrium solution.
Common Traps✓ Known #67
BC only · Euler trap: which point gives the slope?
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Answer#67
The current point \((x_n,y_n)\).
Trap: Using the next point.
Common Traps✓ Known #68
BC only · Can a logistic population stay above \(L\) forever (\(k>0\))?
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Answer#68
No. If \(P>L\), then \(P'<0\) and \(P\) decreases toward \(L\).
Trap: Assuming bigger \(P\) means bigger growth.
Common Traps✓ Known #69
Slope-field trap: what determines the slope at a point?
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Answer#69
The value of \(f(x,y)\) at that point — not the height \(y\) by itself.
Trap: Reading a slope field like a graph of \(y\).
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