Differential Equations

Determine if the differential equation $y^{\prime}=x e^{y}$ is separable, and if so, write it in the form $h(y) d y=g(x) d x$. NOTE: If the equation is not separable, indicate with the checkbox. $d y=$ $d x$ Not separable.

Four million $E$. coli bacteria are present in a laboratory culture. An antibacterial agent is introduced and the population of bacteria $P(t)$ decreases by half every $6 \mathrm{hr}$. The population can be represented by \[ P(t)=4,000,000\left(\frac{1}{2}\right)^{t / 6} \] Part: $0 / 2$ Part 1 of 2 (a) Convert the given model to an exponential function using base $\boldsymbol{e}$. Round the value of $k$ to five decimal places. \[ P(t)= \]

Between 2006 and 2016, the number of applications for patents, N, grew by about $4.8 \%$ per year. That is, $N^{\prime}(t)=0.048 N(t)$. a) Find the function that satisfies this equation. Assume that $\mathrm{t}=0$ corresponds to 2006 , when approximately 451,000 patent applications were received. b) Estimate the number of patent applications in 2020 . c) Estimate the rate of change in the number of patent applications in 2020.

Verify the existence and uniqueness of solutions for the following differential equation: \(\frac{dy}{dx} = 3y + 4x\)

Find a function $f$ and a number $a$ such that \[ 1+\int_{a}^{x} \frac{f(t)}{t^{3}} d t=6 x^{-1} \] \[ f(x)= \] \[ a= \]

Solve the Bernoulli differential equation \(y' + 2y = y^3\).

A company estimates that its sales will grow continuously at a rate given by the function \[ S^{\prime}(t)=24 e^{t} \] where $S^{\prime}(t)$ is the rate at which sales are increasing, in dollars per day, on day $t$. a) Find the accumulated sales for the first 4 days. b) Find the sales from the 2 nd day through the 5 th day. (This is the integral from 1 to 5 .) a) The accumulated sales for the first 4 days is $\$ \square$. (Round to the nearest cent as needed.)

Solve the given differential equation by using an appropriate substitution. The DE is homogeneous. \[ x d x+(y-2 x) d y=0 \]

Find $\frac{d y}{d x}$ \[ e^{4 x}=\sin (x+2 y) \]

A prototype rocket takes off from a launchpad on the ground and is initially at rest. The rocket has a time of flight of $T$ seconds. The velocity of the rocket, $v \mathrm{~ms}^{-1}$, is given by \[ v(t)=0.5 e^{t} \sin \left(\frac{\pi t}{a}\right) \] where $t$ is the time in seconds for $0 \leq t \leq T$ and $a$ is your assigned value in the table above. (a) The rocket malfunctions and begins to descend towards the ground. 2 Find the time at which the rocket begins its descent. (b) Use an approximation method with a suitable number of subintervals to estimate the 2 total distance travelled by the rocket during the first 10 seconds of flight.