Math: First-Order Linear Differential Equations Worksheet

Question 1

Identify whether the differential equation y' + 3y = 6 is first-order linear. If it is, write P(x) and Q(x) from the standard form y' + P(x)y = Q(x).

Accepted Answer

The equation is first-order linear because it contains y' and y to the first power and has the form y' + P(x)y = Q(x). Here P(x) = 3 and Q(x) = 6.

Question 2

Solve the differential equation y' + 2y = 0.

Accepted Answer

The integrating factor is e^(∫2 dx) = e^(2x). Multiplying gives (e^(2x)y)' = 0, so e^(2x)y = C. Therefore, y = Ce^(-2x).

Question 3

Solve the differential equation y' + 4y = 8.

Accepted Answer

The integrating factor is e^(∫4 dx) = e^(4x). Multiplying gives (e^(4x)y)' = 8e^(4x). Integrating gives e^(4x)y = 2e^(4x) + C, so y = 2 + Ce^(-4x).

Question 4

Solve the initial value problem y' + y = e^x, with y(0) = 3.

Accepted Answer

The integrating factor is e^(∫1 dx) = e^x. Multiplying gives (e^x y)' = e^(2x). Integrating gives e^x y = (1/2)e^(2x) + C, so y = (1/2)e^x + Ce^(-x). Using y(0) = 3 gives 3 = 1/2 + C, so C = 5/2. The solution is y = (1/2)e^x + (5/2)e^(-x).

Question 5

Put the equation 2y' + 6y = 10x into standard linear form, then identify P(x) and Q(x).

Accepted Answer

Divide every term by 2 to get y' + 3y = 5x. Therefore, P(x) = 3 and Q(x) = 5x.

Question 6

Solve the differential equation y' - 3y = 6e^(3x).

Accepted Answer

The integrating factor is e^(∫-3 dx) = e^(-3x). Multiplying gives (e^(-3x)y)' = 6. Integrating gives e^(-3x)y = 6x + C. Therefore, y = e^(3x)(6x + C).

Question 7

A slope field represents the differential equation y' + y = 0. Describe the general shape of its solutions and write the general solution.

Accepted Answer

The solutions are exponential decay curves when C is positive and exponential growth downward when C is negative. Solving y' + y = 0 gives y = Ce^(-x).

Question 8

Solve the differential equation y' + (2/x)y = x^2 for x > 0.

Accepted Answer

The integrating factor is e^(∫2/x dx) = e^(2 ln x) = x^2. Multiplying gives (x^2y)' = x^4. Integrating gives x^2y = x^5/5 + C. Therefore, y = x^3/5 + C/x^2.

Question 9

Solve the initial value problem y' + (1/x)y = 4x, with y(1) = 5 and x > 0.

Accepted Answer

The integrating factor is e^(∫1/x dx) = x. Multiplying gives (xy)' = 4x^2. Integrating gives xy = (4/3)x^3 + C, so y = (4/3)x^2 + C/x. Using y(1) = 5 gives 5 = 4/3 + C, so C = 11/3. The solution is y = (4/3)x^2 + 11/(3x).

Question 10

A tank starts with 100 liters of water containing 20 grams of salt. Brine containing 3 grams of salt per liter enters at 2 liters per minute, and the well-mixed solution leaves at 2 liters per minute. Let S(t) be the grams of salt after t minutes. Write the first-order linear differential equation for S(t), then solve it.

Accepted Answer

The salt enters at 6 grams per minute. The salt leaves at a rate of (2/100)S = S/50 grams per minute. The equation is S' = 6 - S/50, or S' + S/50 = 6. The solution is S = 300 + Ce^(-t/50). Since S(0) = 20, C = -280. Therefore, S(t) = 300 - 280e^(-t/50).

Question 11

The graph of several solutions to y' + 2y = 4 is shown. What horizontal line do all solution curves approach as x increases, and why?

Accepted Answer

All solution curves approach the horizontal line y = 2. The general solution is y = 2 + Ce^(-2x), and the exponential term Ce^(-2x) approaches 0 as x increases.

Question 12

Explain the error in this work: To solve y' + 5y = 10, a student writes the integrating factor as 5e^x and then multiplies the equation by 5e^x.

Accepted Answer

The error is that the integrating factor is not found by multiplying the coefficient 5 by e^x. The correct integrating factor is e^(∫5 dx) = e^(5x). Multiplying by e^(5x) makes the left side (e^(5x)y)'.

Math: First-Order Linear Differential Equations

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