Separable First-Order Differential Equations Worksheet

Question 1

Solve the differential equation dy/dx = 3x^2.

Accepted Answer

Integrating both sides gives y = x^3 + C, where C is an arbitrary constant.

Question 2

Solve the differential equation dy/dx = 2xy.

Accepted Answer

Separate the variables to get (1/y) dy = 2x dx. Integrating gives ln|y| = x^2 + C, so y = Ce^(x^2), where C is an arbitrary constant.

Question 3

Solve the differential equation dy/dx = y/5.

Accepted Answer

Separate the variables to get (1/y) dy = (1/5) dx. Integrating gives ln|y| = x/5 + C, so y = Ce^(x/5), where C is an arbitrary constant.

Question 4

Find the particular solution to dy/dx = 4x with initial condition y(0) = 7.

Accepted Answer

Integrating gives y = 2x^2 + C. Using y(0) = 7 gives C = 7, so the particular solution is y = 2x^2 + 7.

Question 5

Find the particular solution to dy/dx = xy with initial condition y(0) = 3.

Accepted Answer

Separate the variables to get (1/y) dy = x dx. Integrating gives ln|y| = x^2/2 + C, so y = Ce^(x^2/2). Using y(0) = 3 gives C = 3, so y = 3e^(x^2/2).

Question 6

Solve the differential equation dy/dx = x/(y), assuming y is not 0.

Accepted Answer

Separate the variables to get y dy = x dx. Integrating gives y^2/2 = x^2/2 + C. Multiplying by 2 gives y^2 = x^2 + C, where C is an arbitrary constant.

Question 7

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

Accepted Answer

Separate the variables to get 1/(y - 2) dy = (x + 1) dx. Integrating gives ln|y - 2| = x^2/2 + x + C, so y = 2 + Ce^(x^2/2 + x).

Question 8

Find the particular solution to dy/dx = 6x/(y^2) with initial condition y(1) = 2.

Accepted Answer

Separate the variables to get y^2 dy = 6x dx. Integrating gives y^3/3 = 3x^2 + C, so y^3 = 9x^2 + C. Using y(1) = 2 gives 8 = 9 + C, so C = -1. The particular solution is y^3 = 9x^2 - 1, or y = cuberoot(9x^2 - 1).

Question 9

A population P grows at a rate proportional to its size: dP/dt = 0.08P. If P(0) = 500, find P(t).

Accepted Answer

Separate the variables to get (1/P) dP = 0.08 dt. Integrating gives ln|P| = 0.08t + C, so P = Ce^(0.08t). Using P(0) = 500 gives C = 500, so P(t) = 500e^(0.08t).

Question 10

An object cools according to dT/dt = -0.2(T - 70), where T is the temperature in degrees Fahrenheit and 70 is the room temperature. If T(0) = 150, find T(t).

Accepted Answer

Separate the variables to get 1/(T - 70) dT = -0.2 dt. Integrating gives ln|T - 70| = -0.2t + C, so T - 70 = Ce^(-0.2t). Using T(0) = 150 gives C = 80, so T(t) = 70 + 80e^(-0.2t).

Question 11

Determine whether the differential equation dy/dx = x + y is separable. Explain your answer.

Accepted Answer

The equation is not separable because x + y cannot be written as a product of one function of x and one function of y. The variables cannot be separated into the form g(y) dy = f(x) dx.

Question 12

A slope field for dy/dx = y is shown. Describe how the slopes change as y increases, and state the general solution.

Accepted Answer

For dy/dx = y, the slope is larger and positive when y is larger and positive, zero when y = 0, and negative when y is negative. The general solution is y = Ce^x.

Separable First-Order Differential Equations

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