Differential Equations — Quiz 2

Questions in this quiz (10)

1. If $$\frac{dy}{dx}=\frac{x}{\sqrt{16+x^2}}$$ and $$y=3$$ when $$x=0$$, then $$y$$ equals:

   • $$\sqrt{16+x^2}-4$$
   • $$\sqrt{16+x^2}-1$$
   • $$\sqrt{16+x^2}+1$$
   • $$\frac{\sqrt{16+x^2}}{2}$$

2. If $$\frac{dy}{dx}=e^{-y}$$ and $$y=0$$ when $$x=1$$, then:

   • $$y=\ln(x)$$
   • $$y=\ln(1+x)$$
   • $$y=\ln(2x-1)$$
   • $$y=\ln(2-x)$$

3. The general solution of the differential equation $$xdy+ydx=0$$ is a family of:

   • circles
   • hyperbolas
   • parabolas
   • lines

4. The curve that passes through the point $$(1,2)$$ and whose slope at any point $$(x,y)$$ is given by $$\frac{dy}{dx}=\frac{2y}{x}$$ has the equation:

   • $$y=2x$$
   • $$y=2x^2$$
   • $$y=2x^3$$
   • $$y=x^2$$

5. If $$\frac{dy}{dx}=\frac{k}{x}$$, where $$k$$ is a constant, and if $$y=3$$ when $$x=1$$ and $$y=5$$ when $$x=e$$, then when $$x=2$$, $$y$$ equals:

   • $$5$$
   • $$3+\ln2$$
   • $$3+2\ln2$$
   • $$5+\ln2$$

6. The slope field shown represents solutions to a differential equation where the slope depends only on $$x$$. The slopes repeat periodically along the $$x$$-axis and are horizontal when $$x=\frac{\pi}{2}$$ and $$x=\frac{3\pi}{2}$$. The slope field corresponds to which differential equation?

   • $$\frac{dy}{dx}=\sin x$$
   • $$\frac{dy}{dx}=-\sin x$$
   • $$\frac{dy}{dx}=\cos x$$
   • $$\frac{dy}{dx}=-\cos x$$

7. The slope field shown below represents the differential equation on the region:$$[-4,4]\times[-4,4]$$A solution curve passing through the origin has been drawn on the slope field.

   • $$\frac{dy}{dx}=x+y;$$ $$y(0)=0$$
   • $$\frac{dy}{dx}=1+x^2;$$ $$y(0)=0$$
   • $$\frac{dy}{dx}=y^2+1;$$ $$y(0)=0$$
   • $$\frac{dy}{dx}=\frac{1}{1+x^2};$$ $$y(0)=0$$

8. Which function is a possible solution of the slope field shown?

   • $$y=1-\ln x$$
   • $$y=1+\ln x$$
   • $$y=1+e^x$$
   • $$y=1+\tan x$$

9. If $$\frac{ds}{dt}=\cos^2\left(\frac{\pi}{2}s\right)$$, and if $$s=1$$ when $$t=0$$, then when $$s=\frac{3}{2}$$, $$t$$ is equal to:

   • $$\frac{1}{2}$$
   • $$\frac{\pi}{2}$$
   • $$\frac{2}{\pi}$$
   • $$\frac{1}{\pi}$$

10. The population of a town increases continuously at a rate proportional, at any time, to the population at that time. The population triples in $$40$$ years. After $$60$$ years the ratio of the population $$P$$ to the initial population $$P_0$$ is:

    • $$\frac{9}{2}$$
    • $$3^{3/2}$$
    • $$\frac{3}{2}$$
    • $$9$$