differential equations-A2

differential equations-A2

by Juno Hu

Multiple Choice

Given the differential equation dP/Dt=5P

A) Find the general solution for P to the differential equation

B)Find the particular solution for P to the differential equation given P(0)= 418

• A) P=Ce^5t

  B) P= 5e^418t

• A) P=Ce^5t

  B) P= 418e^5t

• A) P=Ce^t

  B) P= 418e^t

• A) P=Ce^10t

  B) P= 418e^10t

Multiple Select

In drawing the slope field for the differential equation , I would place short slope lines of _________ at the points (0,1), (1,-1), and (2,-2)   $$\frac{dy}{dx}=2x-3y$$   Select all that apply

• -3

• -1

• 2

• 5

• 10

Multiple Choice

The following Differential Equation is $$\frac{\text{d}y}{\text{d}x}=\frac{y}{x}$$  

• Separable.

• Non Separable

Multiple Choice

2. Solve the differential equation

𝑑𝑦/𝑑𝑡 =3𝑡²/𝑦 with initial condition

𝑦(2) = 0.

• $$y=\ln\left|15t\right|$$

• $$y=16t^3$$

• $$y\ =\ \sqrt{2t^3-16}$$

• $$y=2t^3-16$$

Multiple Choice

The population of the little town of Scorpion Gulch is now 1000 people. The population is presently growing at about 5% per year.

Find the general solution.

• P=.05e^1000t

• P=1000e^5t

• P=1000e^.05t

• p=lne^1000t

Multiple Choice

Determine the value of "c" that satisfies the differential equation $$\frac{dy}{dx}=\frac{x+1}{y+2}$$  if the curve goes through the point (0, -1).

• 5/2

• -3/2

• -1/2

• 1

Multiple Choice

The following Differential Equation is
$$\frac{\text{d}y}{\text{d}x}=x^2y+x$$  

• Separable.

• Non Separable.

Multiple Choice

Which of the following is the solution to the differential equation $$\frac{dy}{dx}=\frac{x^2}{y}$$  with the initial condition y(3) = -2?

• $$y=-2e^{\left(-9+\frac{x^3}{3}\right)}$$  

• $$y=\sqrt{\frac{2x^3}{3}}$$  

• $$y=\sqrt{\frac{2x^3}{3}-14}$$  

• $$y=-\sqrt{\frac{2x^3}{3}-14}$$  

Multiple Choice

Which choice below represents this slope field?

• dy/dx = 2x

• dy/dx = -x

• dy/dx = yx

• dy/dx = x²

Multiple Choice

Given the differential equation dP/Dt=5P

A) Find the general solution for P to the differential equation

B)Find the particular solution for P to the differential equation given P(0)= 418

• A) P=Ce^5t

  B) P= 5e^418t

• A) P=Ce^5t

  B) P= 418e^5t

• A) P=Ce^t

  B) P= 418e^t

• A) P=Ce^10t

  B) P= 418e^10t

Multiple Choice

The following Differential Equation is
$$x\frac{\text{d}y}{\text{d}x}=yx^2$$  

• Separable.

• Non Separable.

Multiple Choice

Solve the following differential equations:
$$\frac{\text{d}y}{\text{d}x}=\frac{1}{\cos y}$$  

• $$\sin y=1+C$$  

• $$\sin y=x+C$$  

• $$-\sin y=\frac{x^2}{2}+C$$  

• $$\sin y=0+C$$  

Multiple Choice

Y is proportional to the difference of x and z

• y= x-z

• y= k(z-x)

• y= k(x-z)

• y= z-x

Multiple Choice

Solve the following differential equations:
$$\frac{\text{d}y}{\text{d}x}=e^x$$  

• $$1=e^x+C$$  

• $$y=\frac{e^x}{2}+C$$  

• $$y=e^x+C$$  

• $$0=e^x+C$$  

Multiple Choice

The following Differential Equation is
$$\frac{\text{d}y}{\text{d}x}=y^2+xy^2$$  

• Separable.

• Non Separable.

Multiple Choice

Let $$\frac{dy}{dx}=-0.4y$$ and  $$y=5\$$  when  $$x=0$$  .  

What is the solution for $$y$$  ?

• $$y=5e^{-0.4x}$$  

• $$y=-0.4e^{5x}$$  

• $$y=5e^{-\frac{x}{0.4}}$$  

• $$y=0.4e^{\frac{x}{5}}$$  

Multiple Choice

Which equation below represents the slope field?

• dy/dx = x - 2

• dy/dx = 1/2x + 1

• dy/dx =1/2 y - 2

• dy/dx = y + 2

Multiple Choice

Solve the differential equation $$\frac{dy}{dt}=\frac{3t^2}{y}$$   with initial condition 𝑦(2) = 0.

• 𝑦 = ln(15t)

• 𝑦 = 16t³

• 𝑦 = (2𝑡³ − 16)^1/2

• 𝑦 = (2𝑡³ −16)

Multiple Choice

The following Differential Equation is
$$\frac{\text{d}y}{\text{d}x}=\frac{e^y}{e^x}$$  

• Separable.

• Non Separable.

Multiple Choice

dy/dx = 2x/e^2y find the general solution

• y = ln (2x/2 + C)

  2

• y = ln(2x/2)+c

• y = e^2x+c

• y = mx+b

Multiple Choice

Solve the following differential equations:
$$\frac{\text{d}y}{\text{d}x}=\frac{x}{y}$$  

• $$\ln y=\frac{x^2}{2}+C$$  

• $$y=\frac{x^2}{2}+C$$  

• $$x^2=y^2+C$$  

• $$y^2=x^2+2C$$  

Multiple Select

Which functions below will have the same slope field? (choose all that apply)

• y = x² - 3

• y = x² + 4

• y = 2x - 5

• y = 2x²

Multiple Choice

Solve the following differential equations:
$$\frac{\text{d}y}{\text{d}x}=\frac{y}{x}$$  

• $$y=Ax$$  

• $$y^2=\ln x+C$$  

• $$\ln y=\frac{x^2}{2}+C$$  

• $$A=yx$$  

Multiple Choice

2. Solve the differential equation

𝑑𝑦/𝑑𝑡 =3𝑡²/𝑦 with initial condition

𝑦(2) = 0.

• 𝑦 = ln(15t)

• 𝑦 = 16t³

• 𝑦 = (2𝑡³ − 16)^1/2

• 𝑦 = (2𝑡³ −16)

Multiple Choice

Solve the following differential equations:
$$\frac{\text{d}y}{\text{d}x}=\frac{1}{\cos y}$$  

• $$\sin y=1+C$$  

• $$\sin y=x+C$$  

• $$-\sin y=\frac{x^2}{2}+C$$  

• $$\sin y=0+C$$  

Multiple Choice

Which slope field is represented by dy/dx = x²?

Multiple Choice

Solve the following differential equations:
$$\frac{\text{d}y}{\text{d}x}=e^x$$  

• $$1=e^x+C$$  

• $$y=\frac{e^x}{2}+C$$  

• $$y=e^x+C$$  

• $$0=e^x+C$$  

Multiple Choice

dy/dx= x √y

Find the general solution

• √y = ln|4x| + c

• √y = (x)^-1 + c

• 2y^1/2 = ln|x| + c

• 1/2y^1/2 = ln|x| + c

Multiple Choice

Solve the following differential equations:
$$\frac{\text{d}y}{\text{d}x}=3y$$  

• $$y=Ce^{3x}$$  

• $$y=3x+C$$  

• $$\frac{y^2}{2}=3x+C$$  

• $$\ln y=x+C$$  

Multiple Choice

A puppy gains weight, w, at a rate approximately inversely proportional to its age, t, in months.

• A

• B

• C

• D

Multiple Choice

y varies jointly with x and the square of z

• y= kxz²

• y= kx√z

• y= kxz

• kx²z

Multiple Choice

dy/dx = 4x/y.  Suppose y(0)=1
The particular solution is

• B
• C
• D
• E

Multiple Choice

what would the sign of the square root after finding c be given the initial condition g(1)= -5

• positive

• negative

Multiple Choice

Water flows continuously from a large tank at a rate proportional to the amount of water in the tank, modeled by dy/dt = ky.  There was initially 10,000 ft^3 at t=0.  After 4 hours there were 8000 ft^3 remaining.
What is the value of k in the differential equation (calculator)

• -0.050
• -0.056
• -.169
• -.200

Multiple Choice

Dy/dx= tanx+15x2+e^x+1/x

• y= -ln|sinx| + 5x³ + e^x + ln|x| + c

• y= secxtanx + 5x³ + e^x + ln|x| + c

• y= -ln|cosx| + 5x³ + e^x + ln|x| + c

• y= sec²(x) + 5x³ + e^x + ln|x| + c

Multiple Choice

Consider the differential equation dy/dx = x + 2y for which g(x) is the solution.  Which of the following statements is true if the particular solution contains (0,-1)

• g(x) is increasing and concave up
• g(x) is increasing and concave down
• g(x) is decreasing and concave up
• g(x) is decreasing and concave down

Multiple Choice

dp/dy = (4y)^-1find the general solution

• p= 1/5y^5/4 + c

• p= 1/4ln|4y| + c

• p= 1/8y² + c

• p= 1 + c

  4y

Multiple Choice

Y is proportional to the difference of x and z

• y= x-z

• y= k(z-x)

• y= k(x-z)

• y= z-x

Multiple Choice

y is proportional to the product of z and the square root of x

• y= z(x²)

• y= kz(x²)

• y= kx√z

• y=zkx

Multiple Choice

y is proportional to the product of z and the square root of x

• $$y=zx^2$$

• $$y=kzx^2$$

• $$y=kz\sqrt{x}$$

• $$y=zkx$$

Multiple Choice

dy/dx = 4x/y. Suppose y(0)=1

The particular solution is

• B

• C

• D

• E

Multiple Choice

∫dx

• x+c

• 2x+c

• 3x+c

• 4x+c

Multiple Choice

Find the particular solution for y if dy/dx = 2x√y and y = 4 when x = 3.

• 2√y = x² + C

• y = (x² + 25)²/4

• y = (x²/2-5/2)^1/2

• y = x⁴/4

Multiple Choice

Find the mistake if possible:

• The Diff EQ is solved correctly

• Step 1 is incorrect. The separation of variables wasn't done correctly.

• Step 2 is incorrect. They didn't integrate x² correctly.

• Step 3 is incorrect. They didn't take the reciprocal of x³/3+C correctly.

Multiple Choice

Find f(1) given f'(x)

• -12

• 0

• 14

• 20

Multiple Choice

Evaluate:

• 4

• 6

• 8

• 10

Multiple Choice

Find the Particular Solution

• $$y=2+e^{\left(\frac{x^2}{2}+x\right)}$$

• $$y=2e^{\left(\frac{x^2}{2}+x\right)}$$

• $$y=\ln\left|\frac{x^2}{2}+x+1\right|+2$$

• $$y=\ln\left|\frac{x^2}{2}+x+e^2\right|$$

Multiple Choice

Solve the Initial Value Problem

• $$y=-\frac{1}{x}$$

• $$y=-x^2$$

• $$y=\frac{-1}{x+1}$$

• $$y=\frac{1}{x+1}$$

Multiple Select

Of the following, which is a solution to the differential equation, select all that apply:
$$y''-6y'+8y=0$$  

• $$y=2\sin\left(4x\right)$$  

• $$y=3e^{2x}$$  

• $$y=Ce^{4x},\$$  where C is a constant