Question 1

Given the differential equation $$\frac{dy}{dx} = y e$$^{x^2}$$, which of the following is the general solution?

Accepted Answer

Question 2

A biologist models the growth of a bacteria population with $$\frac{dy}{dx} = \frac{y}{x} - 2y. If$$ $$y(1) = 2$$, what is the value of $$y$$ when $$x = 2$$?

Accepted Answer

Question 3

Which integrating factor should be used to solve the linear differential equation $$(2x + 3y + 4)dx - (2x + 3y + 1)dy = 0$$?

Accepted Answer

$$e^{\int \frac{2x + 3y + 1}$${2x + 3y + 4} dy}$$

Question 4

A chemical reaction rate is modeled by $$(3x + y + 2)dx - (3x + y + 5)dy = 0. $$What is the general solution to this equation?

Accepted Answer

$$\ln|3x + y + 2| - \ln|3x + y + 5| = C$$

Question 5

Which of the following is the correct integrating factor for the equation $$(x + y + 2)dx - (x - y - 3)dy = 0$$?

Accepted Answer

$$e^{\int (x - y - 3) dy}$$

Question 6

A medical researcher models drug concentration with $$(3x + 2y - 5)dx - (x + y + 4)dy = 0. $$Which of the following is the correct general solution?

Accepted Answer

$$\ln|3x + 2y - 5| - \ln|x + y + 4| = C$$

Question 7

For the equation $$(x - 3y + 1)dx - (2x + y - 4)dy = 0$$, which substitution would best simplify the equation for integration?

Accepted Answer

$$u = x - y$$

Question 8

A physicist models a system with $$(4x + y - 6)dx + (4x + 3y - 2)dy = 0. $$What is the integrating factor to make this equation exact?

Accepted Answer

$$e^{\int (4x + 3y - 2) dy}$$

Question 9

Which of the following is the general solution to $$(3y - 2x)dx + x dy = 0$$?

Accepted Answer

$$y^2$$ - $$2x^2 = C$

Question 10

A population model is given by $$(y - x^2$ y^2$)dx - x dy = 0. $$Which of the following is the correct general solution?

Accepted Answer

Question 11

Which of the following is the correct solution to $$\int x^7$ y^3$ dx = x^8$ y^3$ + C$$?

Accepted Answer

$$x^8$$ $$y^3 + C$

Question 12

A physicist solves $$\int x^7$ y^2$ dx = \frac{1}{8}x^8$ y^2$ + C. $$Which of the following is the correct answer?

Accepted Answer

$$\frac{1}{8}x^8$ y^2$ + C$$

Question 13

Which of the following is the correct solution to $$\int (3y - 2x)dx + x dy = 0$$?

Accepted Answer

$$y^2$$ - $$2x^2 = C$

Question 14

A chemical engineer models a reaction with $$(x^2$$ $$y^2)dx - x dy = 0$. What is the general solution?

Accepted Answer

$$x^2$$ $$y^2 = C$

Question 15

Which of the following is the correct solution to $$\int x^7$ y^2$ dx = \frac{1}{8}x^8$ y^2$ + C$$?

Accepted Answer

$$\frac{1}{8}x^8$ y^2$ + C$$

Question 16

A population model is given by $$(y - x^2$ y^2$)dx - x dy = 0. $$Which of the following is the correct general solution?

Accepted Answer

$$x^2$$ $$y^2 = C$

Question 17

Which of the following is the correct solution to $$\int x^7$ y^3$ dx = x^8$ y^3$ + C$$?

Accepted Answer

$$x^8$$ $$y^3 + C$

Question 18

A physicist solves $$\int x^7$ y^2$ dx = \frac{1}{8}x^8$ y^2$ + C. $$Which of the following is the correct answer?

Accepted Answer

$$\frac{1}{8}x^8$ y^2$ + C$$

Solving First-Order Differential Equations: Methods and Solutions

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