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Amy Ferguson Moncure

Calculus BC · Unit 7

Writing and Verifying Differential Equations

Turn a rate sentence into an equation, then test a proposed solution by differentiating.

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Key Ideas Easy Medium Hard

\(\dfrac{dP}{dt}=kP\)

rate proportional to the amount

What to Remember

Translate the sentence before you solve

Proportional to the amount itself:

\(\dfrac{dP}{dt}=kP\)

Proportional to the difference from a fixed value \(A\):

\(\dfrac{dy}{dt}=k(A-y)\)

A solution is a function

Differentiate the proposed function and substitute it into the equation. Both sides must match for every \(x\) in the domain, not just at one point.

The constant is part of the family

\(y=Ce^{3x}\) is a family of solutions of \(y'=3y\). An initial condition picks one value of \(C\).

In minus out

When a tank is well mixed, the rate of change of the amount is the rate entering minus the rate leaving.

Level 1

Easy

Write the equation, or test one proposed function.

Example
The population \(P\) changes at a rate proportional to \(P\). Write a differential equation.

Proportional to the amount present means a constant times \(P\).

\(\dfrac{dP}{dt}=kP\)

Answer: \(\dfrac{dP}{dt}=kP\)

1
A cup of soup cools at a rate proportional to the difference \(70-T\), where \(T\) is the temperature in degrees. Write a differential equation.
Show Answer

The stated difference is \(70-T\).

\(\dfrac{dT}{dt}=k(70-T)\)

Answer: \(\dfrac{dT}{dt}=k(70-T)\)

2
Show that \(y=4e^{3x}\) is a solution of \(y'=3y\).
Show Answer

\(y'=12e^{3x}\)

\(3y=3\cdot 4e^{3x}=12e^{3x}\)

The two sides match.

Answer: yes

3
Show that \(y=x^2+5\) is a solution of \(y'=2x\).
Show Answer

\(y'=2x\)

The derivative matches the right side. The constant 5 does not appear in \(y'\).

Answer: yes

4
Which function is not a solution of \(y'=2y\)?
A. \(e^{2x}\)
B. \(5e^{2x}\)
C. \(e^{2x}+1\)
D. \(0\)
Show Answer

The derivative of \(e^{2x}+1\) is \(2e^{2x}\).

\(2y=2e^{2x}+2\), which is not the same.

The other three each satisfy \(y'=2y\).

Answer: C. \(e^{2x}+1\)

5
A medicine amount \(A\) decreases at a rate proportional to the amount present. Write a differential equation, and say what the sign of \(k\) tells you if you write \(dA/dt=kA\).
Show Answer

\(\dfrac{dA}{dt}=kA\)

The amount is decreasing, so \(k\) is negative.

Answer: \(\dfrac{dA}{dt}=kA\) with \(k<0\)

Level 2

Medium

A sum of rates, or a function that almost works.

Example
Show that \(y=\sin(2x)\) is a solution of \(y''+4y=0\).

\(y'=2\cos(2x)\)

\(y''=-4\sin(2x)\)

\(y''+4y=-4\sin(2x)+4\sin(2x)\)

\(0\)

Answer: yes

1
Show that every function \(y=Ce^{x^2}\) satisfies \(y'=2xy\).
Show Answer

\(y'=Ce^{x^2}\cdot 2x\)

\(2xy=2x\cdot Ce^{x^2}\)

The two sides match for every constant \(C\).

Answer: yes, for every C

2
A town’s population grows at a rate proportional to the population, and 20 people move in each year. Write a differential equation for \(P\).
Show Answer

Proportional growth contributes \(kP\).

Moving in contributes 20 per year.

\(\dfrac{dP}{dt}=kP+20\)

Answer: \(\dfrac{dP}{dt}=kP+20\)

3
The rate of change of \(y\) with respect to \(x\) is the product of \(x\) and \(y\). Write the equation.
Show Answer

\(\dfrac{dy}{dx}=xy\)

Answer: \(\dfrac{dy}{dx}=xy\)

4
Does \(y=\cos x\) satisfy \(y'=-y\)?
Show Answer

\(y'=-\sin x\)

\(-y=-\cos x\)

Those are not the same function.

Answer: no

5
Find the particular solution of \(y'=4x^3\) with \(y(1)=3\).
Show Answer

\(y=x^4+C\)

\(y(1)=1+C=3\)

\(C=2\)

\(y=x^4+2\)

Answer: \(y=x^4+2\)

Level 3

Hard

A product solution, a mixing sentence, or an initial point on a family.

Example
Show that \(y=xe^{2x}\) satisfies \(y'=2y+e^{2x}\).

Product rule: \(y'=e^{2x}+x\cdot 2e^{2x}\).

\(y'=e^{2x}(1+2x)\)

\(2y+e^{2x}=2xe^{2x}+e^{2x}\)

\(e^{2x}(2x+1)\)

The two sides match.

Answer: yes

1
A 50-liter tank starts with salt water. Pure water enters at 5 L/min. The well-mixed solution leaves at 5 L/min. Let \(S\) be the kilograms of salt. Write \(dS/dt\).
Show Answer

Salt enters at 0 kg/min.

The concentration leaving is \(S/50\) kg/L.

Salt leaves at \((S/50)\cdot 5\) kg/min.

\(\dfrac{dS}{dt}=-\dfrac{S}{10}\)

Answer: \(\dfrac{dS}{dt}=-\dfrac{S}{10}\)

2
The rate of change of \(W\) is proportional to the product of \(W\) and \((100-W)\). Write the differential equation.
Show Answer

\(\dfrac{dW}{dt}=kW(100-W)\)

Answer: \(\dfrac{dW}{dt}=kW(100-W)\)

3
A solution of \(dy/dx=y\) passes through \((0,2)\). Which function is it?
A. \(y=2e^{x}\)
B. \(y=e^{2x}\)
C. \(y=e^{x}+1\)
D. \(y=2e^{2x}\)
Show Answer

The general solution is \(y=Ce^{x}\).

\(y(0)=C=2\)

\(y=2e^{x}\)

Check \(e^{x}+1\): its derivative is \(e^{x}\), not the function itself.

Answer: A. \(y=2e^{x}\)

4
Both \(y=0\) and \(y=e^{4x}\) satisfy \(y'=4y\). Write one family that includes both.
Show Answer

\(y=Ce^{4x}\)

\(C=0\) gives the zero solution.

\(C=1\) gives \(e^{4x}\).

Answer: \(y=Ce^{4x}\)

5
A student claims \(y=e^{x}+e^{-x}\) satisfies \(y''=y\). Check the claim.
Show Answer

\(y'=e^{x}-e^{-x}\)

\(y''=e^{x}+e^{-x}\)

\(y''=y\)

The claim is true.

Answer: yes

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