Use separation of variables to find the solutions to the differential equations subject to the given initial conditions.

$$\frac{d P}{d t}=0.02 P, \quad P(0)=20$$

Find the general solution to the given differential equation. \

$$y^{\prime \prime}+4 y^{\prime}+4 y=0$$

Match the slope fields in Figure discussed earlier to the corresponding differential equations:

$$y^{\prime}=e^{-x}$$

A population, $P$, in millions, is $1500$ at time $t=0$ and its growth is governed by

$$\frac{d P}{d t}=0.00008 P(1900-P).$$

Use Euler's method with $\Delta t=1$ to estimate $P$ at time $t=1,2,3$.

Match the slope fields in Figure discussed earlier to the corresponding differential equations:

$$y^{\prime}=e^{-i^2}$$

Match the slope fields in Figure discussed earlier to the corresponding differential equations:

$$y^{\prime}=x^2 e^{-i}$$

Each curve in the figure represents the balance in a bank account into which a single deposit was made at time zero. Assuming continuously compounded interest, find:

Two curves representing the same interest rate.

Give $k, L, A$, a formula for $P$ as a function of time $t$, and the time to the peak value of $d P / d t$.

$$\frac{1}{10 P} \frac{d P}{d t}=0.012-0.002 P, \quad P_0=2$$

Determine whether each function is a solution to the differential equation and justify your answer:

$$x\frac{dy}{dx}=4y.$$

(d) $y=7x^4$

Match the graphs in Figure discussed earlier with the given description.

The temperature of a piece of steel heated in a furnace and left outside to cool.

Match the slope fields in Figure discussed earlier to the corresponding differential equations:

$$y^{\prime}=\cos x$$

(Continuation of Problem $26$.) Government agencies wam that exposure to air containing $0.02 \%$ carbon monoxide can lead to headaches and diusiness. How long does it take for the concentration of carbon monoxide in the room in Problem $26$ to reach this level?

Determine whether each function is a solution to the differential equation and justify your answer:

$$x\frac{dy}{dx}=4y.$$

(c ) $y=x^3$

Match the graphs in Figure discussed earlier with the given description.

The speed of a constantly decelerating car.