Differential Equations

At each point on a certain curve, the slope of the curve is

 $3x^2y$ .  If the curve contains the point (0,8), then its equation is...

•  $y=8e^{x^3}$  

•  $y=x^3+8$  

•  $y=e^{x^3}+7$  

•  $y=\ln\left(x+1\right)+8$  

If

 $\frac{dy}{dx}=y\ \sec^2x$  and y = 5 when x = 0, then y = ...

•  $e^{\tan x}+4$  

•  $e^{\tan x}+5$  

•  $5e^{\tan x}$  

•  $\tan x+5$  

If

 $\frac{dy}{dx}=2y^2$  and if y = -1 when x = 1, then when x = 2, y = ...

• -2/3

• -1/3

• 1/3

• 2/3

If

 $\frac{dy}{dx}=\sin x\ \cos^2x$  and if y = 0 when    x = $\frac{\pi}{2}$  , what is the value of y when x = 0?

• -1

• -1/3

• 1/3

• 1

The point (3 , 1/4) is on the graph of y = f(x), and the slope at each point (x , y) on the graph is given by  $\frac{dy}{dx}=y^2\left(6-2x\right)$ 
Find y = f(x) by solving the differential equation, then find the value of y when x = 0.

(a)  

Consider the differential equation  $\frac{dy}{dx}=\frac{3x^2}{e^{2y}}$  .

Find the solution to the differential equation satisfying f(0) = 1/2, then find f(1).

(a)