AP Calculus AB Differential Equations and Slope Fields Worksheet (PDF)

AP Calculus AB math worksheet: Differential Equations and Slope Fields – Maths PDF

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This differential equations worksheet for AP Calculus AB covers differential equations and slope fields (Common Core AP CALC AB 7.1, AP CALC AB 7.2, AP CALC AB 7.3, AP CALC AB 7.4, AP CALC AB 7.5, AP CALC AB 7.6, AP CALC AB 7.7), with a free PDF to print.

These 21 problems practice differential equations, slope fields, Euler’s method, separation of variables and exponential growth and decay (AP Calculus AB, topics 7.1 to 7.7). The early problems build the basic skill, and each one uses a different function or situation so that no two feel alike.

Later problems use Euler’s method, initial value problems, Newton cooling and real situations such as population growth and continuous interest. Work in pencil, show every step, and check the answer key only after a full attempt.

AP Calculus AB · Common Core State Standards: AP CALC AB 7.1, AP CALC AB 7.2, AP CALC AB 7.3, AP CALC AB 7.4, AP CALC AB 7.5, AP CALC AB 7.6, AP CALC AB 7.7

Differential equations worksheet: exercises 1 to 10

Exercise 1 – Verify a solution

Verify that \(y=x^{2}+C\) solves \(\frac{dy}{dx}=2x\), and that \(y=3e^{2x}\) solves \(\frac{dy}{dx}=2y\).

Exercise 2 – Verify a second-order solution

Show that \(y=xe^{x}\) is a solution of \(y^{\prime\prime}-2y^{\prime}+y=0\).

Exercise 3 – Reading a slope field

Slope field of dy/dx = x − y.

The slope field is for \(\frac{dy}{dx}=x-y\). Find the slope at \((1,0)\), \((0,2)\) and \((2,2)\). Where are the segments horizontal?

Exercise 4 – Matching an equation

For \(\frac{dy}{dx}=y\), describe the slope field along the line \(y=0\), for \(y > 0\) and for \(y < 0\). Is the slope the same along a vertical line?

Exercise 5 – Euler, two steps

Use Euler’s method with step \(h=0.1\) to estimate \(y(0.2)\) for \(\frac{dy}{dx}=x+y\), \(y(0)=1\).

Exercise 6 – Euler for y prime equals y

Use Euler’s method with \(h=0.5\) to estimate \(y(1)\) for \(\frac{dy}{dx}=y\), \(y(0)=1\). Compare with the exact value.

Exercise 7 – Euler with a decreasing solution

Euler steps for dy/dx = x − y, y(0) = 2.

Use Euler’s method with \(h=0.25\) to estimate \(y(0.5)\) for \(\frac{dy}{dx}=x-y\), \(y(0)=2\).

Exercise 8 – Separable: exponential

Solve \(\frac{dy}{dx}=2xy\) for \(y > 0\).

Exercise 9 – Separable with a trigonometric factor

The solution through (π/2, e).

Solve \(\frac{dy}{dx}=y\cos x\) with \(y\left(\frac{\pi}{2}\right)=e\).

Exercise 10 – Separable: y squared

Solve \(\frac{dy}{dx}=y^{2}\) with \(y(0)=1\), and state the interval on which the solution exists.

Exercises 11 and up

Exercise 11 – Separable: x squared times y

Solve \(\frac{dy}{dx}=3x^{2}y\) with \(y(0)=5\).

Exercise 12 – Separable: a clever substitution

Solve \(\frac{dy}{dx}=e^{x-y}\) with \(y(0)=0\).

Exercise 13 – Separable: y over x

Solve \(\frac{dy}{dx}=\frac{y}{x}\) with \(y(1)=3\) for \(x > 0\).

Exercise 14 – Decay equation

Solve \(\frac{dy}{dx}=-2y\) with \(y(0)=4\), and find \(y(1)\).

Exercise 15 – Population growth

A town’s population over 20 years.

A town’s population grows at a rate proportional to its size: \(\frac{dP}{dt}=0.03P\), \(P(0)=500\) with \(t\) in years. Find \(P(t)\) and \(P(10)\).

Exercise 16 – Doubling time

A bacteria culture doubles every 5 hours and satisfies \(\frac{dN}{dt}=kN\). Find \(k\).

Exercise 17 – Half-life

A medicine decays so that \(\frac{dA}{dt}=-kA\) with a half-life of 10 days. What fraction remains after 30 days?

Exercise 18 – Newton cooling

Cooling toward 70 °F.

A cup of tea at 190 °F cools in a 70 °F room according to \(\frac{dT}{dt}=-0.1(T-70)\). Find \(T(t)\) and \(T(10)\).

Exercise 19 – Continuous interest

An account earns continuously compounded interest at \(\frac{dA}{dt}=0.05A\) with \(A(0)=2000\) dollars. Find \(A(10)\).

Exercise 20 – General and particular solution

Solve \(\frac{dy}{dx}=x(y-1)\), then find the solution with \(y(0)=3\).

Exercise 21 – General exponential model

Show that every solution of \(\frac{dy}{dt}=ky\) has the form \(y=y_{0}e^{kt}\) where \(y_{0}=y(0)\).

Key facts behind this differential equations worksheet

Key facts

Verify a solution by substituting y and y′ into the equation.
A slope field shows dy/dx at grid points; solution curves follow the segments.
Euler: y_new = y_old + h · (dy/dx at the old point), x_new = x_old + h.
dy/dt = ky has the general solution y = Ce^(kt); growth if k is positive, decay if k is negative.

How to use this differential equations worksheet

Work through the exercises in order: they go from simple to challenging. Write every step on paper, then compare with the solutions page and redo any exercise you missed the next day.

Download the differential equations worksheet as a PDF with the button below and keep it with your notes.

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