Find a particular solution, satisfying the initial condition, of each of the following differential equations.

$(y+7) d x+(2 x+y+3) d y=0, y(0)=1$.

Verify that the function satisfies the differential equation. Function: y = 3 cos x + sin x Differential Equation: y″ + y = 0

Verify the solution of the differential equation.

Solution: y=Ce$^{4 x}$, Differential Equation: dy/dx=4y

Find the measures of variability and any outliers for each set of data. the number of minutes spent reading each night: 31, 33, 32, 34, 35, 33

The three air carts shown in Figure 9-16 have masses, reading from left to right, of $4 m, 2 m$, and $m$, respectively. The most massive cart has an initial speed of $v_0$; the other two carts are at rest initially. All carts are equipped with spring bumpers that give elastic collisions. Find the final speed of each cart.(Assume the air track is long enough to accommodate all collisions.)

Use the methods of this section to solve the following differential equations. Compare computer solutions and reconcile differences. $y y^{\prime}-2 y^{2} \cot x=\sin x \cos x$

Consider the differential equation $y^{\prime}(t)+9 y(t)=10$. a. How many arbitrary constants appear in the general solution of the differential equation? b. Is the differential equation linear or nonlinear?

Coach Everly wants to order uniform shirts for all the players p on her women's lacrosse team. Each shirt costs $20. There is an additional cost d for a player to put her name on the shirt. Use the expression to write an equation that can be used to find the total cost if every player on the team orders a shirt with her name on it. number of players (cost of shirt + cost of name) = total cost

Consider the differential equation $y^{\prime}(t)+9 y(t)=10$. Is the differential equation linear or nonlinear?

Populations of aphids and ladybugs are modeled by the equations

$$\dfrac{dA}{dt} = 2A - 0.01AL$$

$$\dfrac{dL}{dt} = -0.5L + 0.0001AL$$

The direction field for the differential equation in the previous part is shown. Use it to sketch a phase portrait. What do the phase trajectories have in common?

What is a differential? What is it used for?

The suction and discharge pipes of a pump are connected to a differential mercury manometer. The diameter of the suction pipe is $200 \mathrm{~mm}$ and the diameter of the discharge pipe is $150 \mathrm{~mm}$. The specific gravity of the pumped liquid is $S=0.81$, and the manometer reading is $H=600 \mathrm{~mm}$. Determine the pressure head rise across the pump if the discharge is $Q=115 \mathrm{~L} / \mathrm{s}$. The centerlines of the suction and discharge pipes are at the same elevation.

Consider the differential equation y'(t)+9y(t)=10. a. How many arbitrary constants appear in the general solution of the differential equation? b. Is the differential equation linear or nonlinear?

A high-speed subsonic airplane is flying at Mach 0.65. A Pitot tube on the wing tip measures a pressure of 2339 lb/ft2. What is the altitude reading on the altimeter?