13. Intro to Differential Equations

State the order of the differential equation and indicate if it is linear or nonlinear. 

${\left(y^{\prime \prime }\right)}^2+6e^t{y}^{\prime }=4t$

State the order of the differential equation and indicate if it is linear or nonlinear. 

$y^{\prime }\bullet y=3e^t$

Find the particular solution to the differential equation $y^{\prime }=2e^t+4t$ given the initial condition $y\left(0\right)=1$.

Find the general solution to the differential equation $\frac{dy}{dx}=-2x+5x^2$.

Find the particular solution to the differential equation $y^{\prime }=2\sin x+3\cos x$ given the initial condition $y\left(0\right)=4$.

43–44. Motion in a gravitational field: An object is fired vertically upward with initial velocity v(0)=v₀ from initial position s(0)=s₀.

a. For the following values of v₀ and s₀, find the position and velocity functions for all times at which the object is above the ground (s = 0).

v₀ = 49 m/s, s₀ = 60 m

45–46. Harvesting problems Consider the harvesting problem in Example 6.

If r = 0.05 and H = 500, for what values of p₀ is the amount of the resource decreasing? For what value of p₀ is the amount of the resource constant? If p₀ = 9000, when does the resource vanish?

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. The general solution of the differential equation y'(t) = 1 is y(t) = t

Explain how the growth rate function determines the solution of a population model.

Explain how a stirred tank reaction works.

What are the assumptions underlying the predator-prey model discussed in this section?

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. The differential equation y′+2y=t is first-order, linear, and separable.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

d. The direction field for the differential equation y′(t)=t+y(t) is plotted in the ty-plane.

7–16. Verifying general solutions Verify that the given function is a solution of the differential equation that follows it. Assume C, C1, C2 and C3 are arbitrary constants.

y(t) = C₁ sin4t + C₂ cos4t; y''(t) + 16y(t) = 0

7–16. Verifying general solutions Verify that the given function is a solution of the differential equation that follows it. Assume C, C1, C2 and C3 are arbitrary constants.

u(t) = C₁eᵗ + C₂teᵗ; u''(t) - 2u'(t) + u(t) = 0