Question

1. what is the order of the differential equation $\frac{d^{3}y}{dt^{3}}+(x\frac{dy}{dx})^{2}=y^{4}$

(a) the order is 1.
(b) the order is 2.
(c) the order is 3.
(d) the order is 4.

1. we wish to determine how much the height of a 25 - year - old tree will increase in the next year. if $h(t)$ gives the height as a function of age in years, which of the following gives $delta h$? (select all that apply).

(a) $int_{0}^{1}h(t)dt$
(b) $int_{0}^{1}h(t)dt$
(c) $int_{25}^{26}h(t)dt$

Explanation:

Step1: Recall order of differential equation

The order of a differential equation is the highest - order derivative present in the equation.

Step2: Identify highest - order derivative

In the given differential equation $\frac{d^{3}y}{dt^{3}}+(x\frac{dy}{dx})^{2}=y^{4}$, the highest - order derivative is $\frac{d^{3}y}{dt^{3}}$, which is a third - order derivative.

for question 5:

Step1: Recall the fundamental theorem of calculus

The change in a function $F(x)$ over an interval $[a,b]$ is given by $F(b)-F(a)=\int_{a}^{b}F^{\prime}(x)dx$. Here, we want to find the change in the height function $\Delta H$ over the interval from $t$ (age of the tree) to $t + 1$ (one - year later). If we assume the current age of the tree is $t$ and we want to find the change in height in the next year, and $H(t)$ is the height function and $H^{\prime}(t)$ is the rate of change of height with respect to time.

Step2: Determine the correct integral

The change in height $\Delta H$ over the interval $[t,t + 1]$ (a one - year interval) is given by $\int_{0}^{1}H^{\prime}(t)dt$ according to the fundamental theorem of calculus.

Answer:

c. The order is 3.