Calc AB Unit 7 Differential Equations

Separation of Variables

100

Determine whether the differential equation is separable. If so, write the separated form.

dy/dx = x / (1 + y²)

(1 + y²)dy = xdx

200

Solve the differential equation for dy/dx = 3x/y

y²/2 = 3x²/2 + C

300

Solve the differential equation dy/dx = y/x

given y(2) = 6.

y = 3x

400

Solve the differential equation dy/dx = xy²

given y(0) = 1

-1 = C

500

Solve the differential equation dy/dx = 2y/(1 + x²).

Ce^2arctan(x)

AP Mistakes

100

A student separates variables for dy/dx = x/y

and writes: dy = x/y dx

What is the mistake and what is the correct answer?

They did not separate the variables.

ydy = xdx

200

A student solves dy/dx = 2x and writes the solution 

y = x²

What mistake did they make?

They forgot the constant of integration (+ C).

300

A student solves dy/dx = xy

and gets ln y = x² + C.

What mistake did they make, and what is the correct solution?

They integrated incorrectly.

ln |y| = x²/2 + C

400

A student tries to solve dy/dx = 2x/y²

and writes the solution as y = x² + C

What mistake did they make and what is the correct solution?

They forgot to properly separate variables and integrate y^-2.

Correct separation: y² dy = 2x dx

Integrate: y³/3 = x² + C

500

A student solves the differential equation

dy/dx = y(2−y) and writes the solution as

y = 2−Ce^2x. What is the correct solution?

y/(2 - y) = Ce^x 

Particular Solutions

100

Given: y = x² + C

If the curve passes through (1,5), find C.

C = 4

200

The function y satisfies

dy/dx = x² 

and the graph of the solution passes through the point (2,5). Find the particular solution.

C = 7/3

y = x³/3 + 7/3

300

Solve the differential equation

dy/dx = x/y

given y(1) = 2

C = 3

y² = x² + 3

400

The solution to the differential equation 

dy/dx = y(1 - x) passes through (0,4).

Find the particular solution.

4 = C

500

A function y satisfies the differential equation

dy/dx = x²/y

The graph of the solution passes through the point (1,2). Find the particular solution.

C = 10/3

y² = 2/3x³ + 10/3

Exponential Growth & Decay

100

Identify the carrying capacity in

dP/dt = 0.4P(1 − P/800).

800

200

A population satisfies

dP/dt = −0.2P

Interpret the meaning of the negative constant.

The population decays exponentially.

300

For dt/dP = kP(1 − P/L)

What happens when P > L?

The population decreases.

400

A population grows according to dP/dt = 0.6P

If P(0) = 100, write the solution.

P = 100e^0.6t

500

A substance decays according to

dA/dt = −kA

Explain why the rate of change decreases as A decreases.

The rate is proportional to the amount present.