Differential Equations — Quiz 1

Questions in this quiz (9)

1. Find the particular solution to $$\frac{dy}{dx}=\frac{\sqrt{y}}{9+x^2}$$ that satisfies the condition $$y(0)=4$$.

   • $$\left(2+\frac{1}{3}\tan^{-1}\left(\frac{x}{3}\right)\right)^2$$
   • $$\left(2+\frac{1}{6}\tan^{-1}\left(\frac{x}{3}\right)\right)^2$$
   • $$\left(2+\frac{1}{2}\tan^{-1}\left(\frac{x}{3}\right)\right)^2$$
   • $$(4+\tan^{-1}\left(\frac{x}{3}\right))^2$$

2. What function has derivative $$f'(x)=e^x\sec x$$ and passes through the origin?

   • $$f(x)=e^x\sec x$$
   • $$f(x)=e^x\tan x$$
   • $$f(x)=\int_0^x e^t\sec t\,dt$$
   • $$f(x)=e^x\sec x-e^0\sec0$$

3. In some cases, the rate of change of a quantity is proportional to the quantity itself. This is written as $$\frac{dy}{dt}=ky$$. What is the general solution?

   • $$y=Cke^t$$
   • $$y=Ck^t$$
   • $$y=Ce^{t/k}$$
   • $$y=Ce^{kt}$$

4. A particle moves along the x-axis with acceleration at time $$t$$ given by $$a(t)=4t$$. Find the function describing the particle's position if it travels at a rate of $$1$$ ft/sec when $$t=0$$ and is $$2$$ feet to the right of the origin when $$t=1$$.

   • $$s(t)=\frac{2}{3}t^3+t+\frac{1}{3}$$
   • $$s(t)=\frac{2}{3}t^3+t$$
   • $$s(t)=\frac{4}{3}t^3+t$$
   • $$s(t)=\frac{2}{3}t^3+t-\frac{1}{3}$$

5. If a ball is thrown upward and reaches its maximum height when $$t=3$$ seconds, and if the ball is $$120$$ feet off the ground when $$t=2$$ seconds, what is the position function for the ball?

   • $$s(t)=-16(t-3)^2+152$$
   • $$s(t)=-16(t-3)^2+136$$
   • $$s(t)=-16(t-3)^2+120$$
   • $$s(t)=-32(t-3)^2+136$$

6. Use Euler's Method to approximate $$y(2)$$ if $$\frac{dy}{dx}=\frac{x}{y}$$ and $$y(1)=2$$ using one step of size $$h=1$$.

   • $$2.25$$
   • $$3$$
   • $$2.5$$
   • $$2.75$$

7. Use Euler's Method to approximate $$y(4)$$ for $$\frac{dy}{dx}=\frac{2}{x}$$ given that $$y(3)=1$$ using one step.

   • $$\frac{7}{3}$$
   • $$2$$
   • $$\frac{4}{3}$$
   • $$\frac{5}{3}$$

8. A culture of bacteria grows at a rate proportional to the number present. Initially, there are $$20$$ bacteria. After $$10$$ minutes, the population grows to $$60$$ bacteria. How many bacteria will there be after $$30$$ minutes?

   • $$180$$
   • $$540$$
   • $$120$$
   • $$360$$

9. A radioactive isotope has a half-life of $$20$$ days. If a sample initially has a mass of $$5$$ kilograms, how long will it take for the sample to decay to $$200$$ grams?

   • $$86\text{ days}$$
   • $$92\text{ days}$$
   • $$100\text{ days}$$
   • $$110\text{ days}$$